I already wrote a piece about the process of disputing YouTube copyright claims, but after gaining some more experience, I realized that I was less bothered about what to do and more bothered about what evil will befall me if I do it.  So this comes to answer that question.

The most important thing about dealing with YouTube copyright claims is to get over the fear barrier.

YouTube will not take revenge on you and will not sue you.  The music companies will not sue you.  They can, but they won't.

If worse comes to worse and they do: (a) you'll get help and (b) you can always drop it before things get bad.

The process of dealing with copyright claims:

block -> dispute -> block again -> appeal -> copyright strike -> counter notification

(copyright owner actions in red, your options in green)
Disputing and appealing unblocks your vid, until the copyright-holders respond.

What to do: Dispute or file a counter-notification.

The video companies will not chase you and will not respond at all, thereby releasing their claim.  Once they drop it, your vid is clear.

What to do:

If you vid is blocked, dispute it. The vid will be restored and the music companies have a month to respond.

Some will respond immediately, some will wait the whole month, but at some point, they'll respond and your vid will be blocked again.

Once your vid is blocked again, decide if it's worth your while to appeal.  Once again, the vid will be restored and the music companies have a month to respond.

Take into account that the appeal will most likely end with a copyright strike and your vid will be removed for at leat 3 weeks.  You need to decide whether it's worth the hassle.  Your vid might be blocked in only one country (ie, Germany), but if it gets taken down for a few weeks, it won't be accessible at all.

It is very rare for an appeal to end with releasing the claim (I had only one such case, with WMG).

If you get a copyright strike, post a counter-notification. After three weeks, your vid will be restored.

Until the next time the music companies decides to bug you about it.

It will start a cycle: They block you, you appeal, your vid is blocked for a few weeks only to be released later.
However, the more cycles you go through, the more you risk your vid being taken down permanently.  That is, you won't be able to appeal that vid anymore.

I've been told this is especailly true of UMG.  They have a deal with YouTube and if you appeal too much, the vid gets taken down.  So it's really a question of how much you want your vid unblocked.

However, if you don't mind the hassle or fear that your vid will be taken down - in some cases, the music companies give up at some point.

The whole process takes a very, very long time.  You can dispute as much as you want, but you can only appeal three vids at a time.  You will only get one strike if/when the music companies decide to reject your appeal.  But since you can only appeal if you don't have a copyright strike, you might need to wait (at least) two months between every three appeals.

YouTube might update you about the status of your dispute or appeal, but I have found that these emails do not necessarily reflect reality.