As I was reading Isaiah 61 & 62,  God brought to mind verse after verse that supported or further explained what He was saying in these chapters.  First I would suggest that you read the two chapters in your Bible and then the chapters with my speculations and references interwoven.  I hope it helps to give you a bigger picture of Isaiah 61 & 62.
     These chapters describe something that I believe God has laid on my heart to do and be.  I've read it in the past, but wasn't hit with the significance of it until just recently. 

Comments on: 61 & 62

My Comments on Isaiah 61

The Spirit of the Sovereign LORD is on me,

He's on me and in me, and sometimes He overwhelms me,  
and He's the same spirit that is on Jesus .
(Acts 1:8)(Lk 4:18-19)

because the LORD has anointed me to preach good news to the poor.

He's called each of us to preach good news to the poor .  This is what Jesus did and what He's commissioned us to do as well . The anointing is the Holy Spirit.  That physical oil that the Old Testament talks about is the anointing of the Holy Spirit that I John 2:27 talks about.  The Holy Spirit has put a desire in me to tell the poor (financially, spiritually, emotionally, socially, and mentally poor) all about God's provision, His healing, power and love!  
(James 1:27)(Jn 17:18)(Mt 11:5)

He has sent me to bind up the brokenhearted,

There are brokenhearted are everywhere.  I don't think I know anyone who hasn't had their heart broken.  And many of the people I know are suffering from a broken heart right now.  I want them to know the love and warmth of God.  I want to hold them in hugs so sincere that they can cry it out and in the end feel better; not just better, but really know in their spirits that it's going to be ok because God loves them and God knows. 
When I was a kid I cried myself to sleep many nights.  All I wanted was someone to hold me and comfort me.  I prayed for that someone, a best friend, a lover, or a daddy, and for a long time no one came.  Eventually God told me that He was my daddy, and I could climb up into His lap and He'd hold me, let me cry, and kiss my tears away.  Now I'm married.  It took me a while to be brave enough to cry on my husband, but now I see that God heard me.  He was preparing someone who would be able to physically hold me, comfort me, and kiss my tears away.  I think one of the reasons it took a while for God to give me that person is because I had to know His comfort first .  Once I knew that He was my comforter I was able to accept comfort from my husband.  
(II Cor 1:3)(Jn 14:16)(Lk 4:18)

to proclaim freedom for the captives and release from darkness for the prisoners,

I've been thinking about this verse more as I prepare for the prison ministry.  Not all prisoners can be freed from their physical chains just yet, but all prisoners can be released from darkness. That's what gets many of them into prison in the first place.  The prince of this world is a liar, and his realm is a realm of spiritual darkness.  When people are living under his authority their lives are incredibly dark.  I want to see the prisoners of darkness set free.  
There's a prison in Argentina where so many of the inmates have met God that there are very few left in the entire prison who live in darkness.  They are praying twenty-four hours a day, every day, taking turns to pray for their prison, families, country, and get this.... they're praying for the salvation and revival of America.  Inmates in Argentina!  Praise God!  
(Lk 1:79)(Jn 1:5)(Jn 12:31)(Eph 2:2)(Jn 8:44)
(II Cor 1:10)(Gal 3:23-24)(Acts 12:7)

to proclaim the year of the LORD's favor and the day of vengeance of our God,

The year of the LORD's favor isn't something to be taken lightly.  God's favor is amazing, remember Obed-Edom who ended up with the Ark of the Covenant at his house?  That was favor, that was amazing.  And his vengeance?  That's even better, Eph 6:12 tells us that we aren't fighting flesh and blood (even if some of us are, they aren't the real enemy, they're distractions)  The real enemy is Satan, and our own flesh and selves.  That's why God's vengeance is so wonderful, because He kills the enemy, there is no more temptation, confusion, sickness, oppression, depression, ....the list goes on and on.  God gives His favor, and part of that is giving us freedom from our enemies.  
(Pr 19:12)(Pr 8:35)(II Sam 6)(Deut 32:41)(Ps 94:1)(Rm 12:19)

to comfort all who mourn,

God gave us the Holy Spirit to comfort us.  God is the master comforter, He always knows what's wrong, and always knows how to fix it .  He's never too busy or too tired.  He's always there for us.  The greatest part is that when He comforts us, it isn't a quick fix, that will make us feel better for a little while, but when He comforts us, we know deep down where it hurts, that it's going to be alright because He is our provider and our protector .  Everything we need is found in Him.  That's amazing! (Mt 5:4)(Mt 6:8)(Mt 6:32-33)

and provide for those who grieve in Zion-

Lately Steve and I (Heidi) have been learning a lot about God's provision.  We haven't been grieving, but we've been in need of His provision in many other areas.  He's faithful .  He does provide for your every need (physical, social, emotional, spiritual, mental, everything).  Also, the "in Zion" I think signifies those who belong to Him.  I don't know if  He provides for everyone's needs, but I know that He provides for the needs of His children .  
(Gen 22:14)(Lam 3:32)(Phil 4:19)

to bestow on them a crown of beauty instead of ashes,
the oil of gladness instead of mourning,
and a garment of praise instead of a spirit of despair.

This is something Jesus did for people.  He was often healing people, raising people from the dead, and doing all kinds of miracles.  I usually think of the people who get healed or raised from the dead, but what about their friends and relatives, the people who were mourning for the dead? Once God raised the dead, those people's mourning was turned to gladness. He does it in our lives through the Holy Spirit.  There have been times when I've been praying, just heavy with a spirit of despair, and God has turned that into praise. It's amazing to see the change that being in His presence makes.  This is what I want to do for others too.  I want to show them the love and hope that God has given us.  I want them to have a spirit of praise instead of despair .  I'm not sure about the "crown of beauty instead of ashes" part.  As far as I know, the Jews put ashes on their heads when they were morning or when something awful had happened and they needed to seek God's help.  At the same time they wore "sackcloth" I'm not sure what that is either, except that it was the standard dress for a time of mourning.  Certainly, it would be much more pleasant to have a beautiful crown on your head than ashes. 
(Lk 8:52)(Esther 9:22)(Heb 1:9)(Ps 30:11-12)(Acts 16:24-25)(Ps 30:11)

They will be called oaks of righteousness,
a planting of the LORD for the display of his splendor.

This reminds me of the blind man in John 9.  The people asked Jesus who had sinned and caused the blindness, whether it was his parents or himself.  Jesus said that neither had sinned, but that it happened so that Jesus could be glorified.  Also, in Ezekiel (47:12), God talks about His people as being planted by streams of living water.  He's talking about the people from the earlier verses, the brokenhearted, the captives, the poor.  He's taking them from their situations, giving them life , and life abundantly, and now, calling them oaks of righteousness to display His splendor.  It's a good deal!  I know he's done this for me, and I want to see it happen for others.  
(Jn 10:10)

They will rebuild the ancient ruins and restore the places long devastated;
they will renew the ruined cities that have been devastated for generations.

This seems to be the time lost when people are not free from sin or darkness.  While people are waylaid being slaves to the Prince of this world things fall apart.  Life gets ruined and devastated.  It needs to be restored, and God is a master builder.  He's a master re-builder.  See (Acts 22) the examples from Saul/Paul's life. And in Christ's example of the Cross.  We are the temple of the Holy Spirit, and we've been devastated for so long.  Finally, God did two things at once, he rebuilt His temple, Jesus, and his temple, the body of Christ, the believers.
(Eph 2:2)(Jn 2:19)(I Cor 3:16)

Aliens will shepherd your flocks;
foreigners will work your fields and vineyards.

I think this bit is mainly saying that God will prosper us.  One way to get foreigners to work for you (if you're an Israelite back in the day) is to go to battle with another country and win.  Once you've won you get to take slaves and animals. So this can translate into God giving us victory in battle.  This is something we desperately need, not only against physical enemies of war, but against the spiritual enemies, which are the true enemies and the much more dangerous sort.  But God has gained the victory! and has given us authority, and He will provide for us on every front.  
(I Jn 5:4)(I Cor 15:57)(Lk 10:19)(Mt 10:5-10)((Mt 6:32)

And you will be called priests of the LORD,
you will be named ministers of our God.

I went to a Catholic high school, and this was not the accepted opinion.  But, since God has presented it to us in scripture I believe Him.  Not only are we priests, but we can come in confidence  to him.  When Christ died, the veil was torn in two.  The veil used to separate the area in the temple where God's presence was from the area when all the people were allowed to go to worship God.  Now, we have free access to God any time we want, we can come to Him and talk with Him, listen to Him and bring others into His presence.   
(I Pet 2:9)(Eph 3:12)(Ex 26:33)(Lk 23:45)(II Cor 5:20)

You will feed on the wealth of nations,
and in their riches you will boast.

Once again, the beauty of God's provision for us.
(Phil 4:19)

Instead of their shame my people will receive a double portion,

Remember Job?  He was shamed; everything was taken from him and he was almost completely destroyed.  He got mainly bad advice from his friends, his troubles just increased and intensified, but God, even though he was allowing it to happen to him, God knew his heart.  God knew that Job wouldn't turn from him.  He knew that Job was faithful.  And because of God's mercy and Job's steadfastness, in the end he received a double portion.
(Job 2:9-11)(Job 1:8-12)(Job 42:12)

and instead of disgrace they will rejoice in their inheritance;
and so they will inherit a double portion in their land,
and everlasting joy will be theirs.

God has given us an inheritance with Jesus.  Since we are the body of Christ and the bride of Christ, we inherit with Him whatever he inherits.  In John (17:21), Jesus says, "I pray...that they may be one, Father, just as you are in me and I am in you.  May they also be in us so that the world may believe that you have sent me."  That's huge!  
I had a vision once that I was walking with God in His house.  It was beautiful, huge, and there were amazing things in it.  In our life, my husband and I had been going through a really tight place financially, and so I was trying to figure out how to ask God to provide for us.  I didn't want to seem forward about it, because it seemed such an amazing thing that I could just walk with Him in His house, but I said something about it, and He just put His arm around me and said, "You know, it's all yours."  That was so comforting.  One of those times when it's so comforting that it makes you cry.  
Remember the prodigal son? I mean the one that stayed home.  When the brother came back he was so mad.  He wanted the attention and gifts and party that the Father gave his brother.  When confronted about it, the Father said, "My son, you are always with me, and everything I have is yours."  That's our inheritance.  We have it, He already died.  The great thing about it is that He died, we inherited so much with Jesus, and he came back to life, He's with us now, and we still have this wonderful inheritance.  
(Rm 8:17)(I Cor 12:27)(Eph 5:31-32)(Rev 21:2, 9)(Jn 3:29)(Lk 15:11-31)

"For I, the LORD, love justice;
I hate robbery and iniquity.
In my faithfulness I will reward them and make an everlasting covenant with them.
Their descendants will be known among the nations and their offspring among the peoples.
All who see them will acknowledge that they are a people the LORD has blessed."

To go from someone trapped in darkness, mourning, grief, despair (the whole first few verses of this chapter) to people that are set free, rewarded, and in a covenant with God is amazing.  Not only that, but people who see them and their families will know that it is God who has blessed them.  He works it all together for His glory.

I delight greatly in the LORD;
My soul rejoices in my God.
For he has clothed me with garments of salvation and arrayed me in a robe of righteousness,
as a bridegroom adorns his head like a priest,
and as a bride adorns herself with her jewels.

Steve and I got married summer 2001, and that delight and rejoicing, getting dressed up in beautiful clothes and jewelry, that's only an itty bitty picture of what God's talking about here.  
Our delight in God is to be so much bigger than that.  We're actually the bride of Christ.  Married to the God of the entire universe.  (Nope, it's not just the nuns)  So many places He gives the picture of our relationship with Him as the bride and groom.  That's the love and excitement He wants us to feel for him, that's the commitment He wants us to have to him.  That's part of the covenant He's talking about.  Not only that but the clothes we wear, our spirit's wedding clothes,  are salvation and righteousness.
(Hosea 2:14-23)(Rev 19:7-8)

For as the soil makes the sprout come up and a garden causes seeds to grow,
so the Sovereign LORD will make righteousness and praise spring up before all nations.

Amazing! Look at the mustard seed and the seed the farmer threw.  His word never comes back void.
(Mt 17:20)(Mt 13)(Is 55:11)

My Comments on Isaiah 62

For Zion's sake I will not keep silent,
for Jerusalem's sake I will not remain quiet,
till her righteousness shines out like the dawn,
her salvation like a blazing torch.

Isaiah was a prophet, which means that he often had to tell people things "for their own good".  As you can well imagine people don't often like to hear from prophets how God wants them to change so that they will be more righteous.  Even so, this was one of Isaiah's commissions.  He was to speak the words God gave him, to tell Zion and Jerusalem (God's people) whatever it was God wanted them to know, and he was to tell them until their righteousness shined out.  If we listen and take cover in God against our attackers, then we will be victorious, rather, God will be victorious in us.  If we heed His words to us, and we allow ourselves to be molded and fired then we will shine in His glory and the righteousness we have through Christ.  
(Ps 32:7)(Is 64:8)(Jer 18:4-6)(Zec 13:9)

The nations will see your glory;
you will be called by a new name that the mouth of the LORD will bestow.

This is something that I've been looking forward to for a long time.  Sometimes I try to guess my new name, but I know that won't work, I could never guess it.  But I'm thinking about names that have meanings.  In the Bible the names people were given at birth had a lot of significance, and when God changed the person He often changed their name too.  Sarai/Sarah, Saul/Paul, Jacob/Israel....  There are a lot of people who have been called a lot of bad names, they've been called worthless, hopeless, ugly, evil, scary, bad, mean, pathetic,  and the list goes on.  He wants to change our names.  
(Gen 17:15)(Acts 9)(Gen 32:28)

You will be a crown of splendor in the LORD's hand,
a royal diadem in the hand of your God.
No longer will they call you Deserted,
or name your land Desolate.
But you will be called Hephzibah, (my delight is in her)
and your land Beulah; (married)

He wants to change our names to Mrs. Christ.  The bride of Christ,  Christian.  He wants to change our names to Love, Friend, Daughter, Son, Beloved, Wife, Child, Accepted, and more.

for the LORD will take delight in you,
and your land will be married.
As a young man marries a maiden,
so will your sons marry you;
as a bridegroom rejoices over his bride,
so will your God rejoice over you.

Like I said before about Steve and I, this is one I didn't really understand until we were married.  But now I know what it's like for someone to take delight in me, and I know that God delights in us so much more than any husband could delight in his wife.  God's love is so much more than anything we can imagine. 
(Jn 3:27-36)

I have posted watchmen on your walls, O Jerusalem;
they will never be silent day or night.

Like in Ezekiel 33, the watchman had to go tell the people when an attack was coming on, or when God had a message for them.

You who call on the LORD give yourselves no rest,
and give him no rest till he establishes Jerusalem and makes her the praise of the earth.

There you go, another pray without ceasing, and be persistent.  God is so serious about us, He wants to hear that we are serious about Him.  He wants us to know that we're serious.  Call on Him and He will answer you! 
(I Thes 5:17)(I Sam 12:23)(Lk 11:5-10)(Mt 15:21-28)

The LORD has sworn by his right hand and by his mighty arm:
"Never again will I give your grain as food for your enemies,
and never again will foreigners drink the new wine for which you have toiled;
but those who harvest it will eat it and praise the LORD,
and those who gather the grapes will drink it in the courts of my sanctuary."

 I wonder if the grain can be the grain from that harvest in the NT.  So many times I've seen people come to God, or almost come to God, but then get stolen away from Him.  I want to see them remain in Him.
(Mt 9:35-38)(Jn 15:1-17)

Pass through, pass through the gates!
Prepare the way for the people.
Build up, build up the highway!
Remove the stones.
Raise a banner for the nations.

Like John did, there is still much preparation to be done!
(Mk 1:2-8) 

The LORD has made proclamation to the ends of the earth:
"Say to the Daughter of Zion,
'See, your Savior comes! See, his reward is with him,
and his recompense accompanies him.'"

This is a message of hope.  Many live without any hope at all, and in the middle of a war zone, it's hard to have hope even if you're normally inclined to.  But we are to encourage each other and say, "See, Here he comes!"  He comes, and He is amazing and good.  He'll save us from all the gunk we've lived in, and had poured on us from our enemies.  He'll heal our wounds and bind up our broken hearts.
(Rm 13:10-14)(Eph 5:14-21)(I Thes 5:10-11)

They will be called the Holy People,
the Redeemed of the LORD;
and you will be called Sought After,
the City No Longer Deserted.

Sought after.  Wow.  Not many people feel sought after, unless it's the law that's seeking after them.  But God wants us.  He's looking for people to worship Him to be his children.  He wants to set us free and give us His life and His peace.  We aren't deserted. Ever.
(Song of Solomon 7:10)(Ps 53: 2)(Jn 4:23-24)(Rm 8:14-16)(Jn 14:27)(Heb 13:5-6)