Comments:
It's "up to" $10,000 per infringement - the number is not automatic, there are several considerations (such as intent vs. innocent infringement).
Publishers (and labels) often ask for an assignment of a percentage of the copyright - It is also common for well known artists to request a share of the copyright as part of the deal to release the song. You will sometimes find language in publishing deals that say the publisher agrees to register the work in the authors name (or both) read it carefully.
It is common to use an SR form to claim registration for both the sound recording and the song itself (by filling in the nature of the copyright claimed as "words, music and performance).
Don't depend on notarized copies, or witnesses, or mailing it to yourself, or any of the other creative ideas expressed in this post - registration is prima facie evidence of ownership. That means that under the law if somebody else registers it before you do, they are presumed to be the owner - and it's a real uphill battle to try to prove otherwise when they are entlted to the presumption. Register the damn songs!