Absolutely, you need permission.
Even the free stuff available from many Creative Commons websites has some kind of obligation to follow through on - usually a note in the credits, at the minimum.
If you're looking to use "real" known bands, you'll probably need to go through ASCAP or BMI (and pay the appropriate fees). That's assuming the song is even available, for starters. Just 'cos you might be real-life pals with Steven Tyler, doesn't mean he can just give you permission to use any Aerosmith song you want. They can be restricted by labels, or various contracts.
It's much easier to get permission to use songs from bands found on MySpace, or similar sites. A few e-mails back & forth, and a quick talk about money issues (heck, if it's none just be upfront about it), and get them to sign a release for you. You'll need that release, if you ever submit to a festival that plays by the rules, or look to get your flick picked up by anyone.
All that's going to cost you is less than a buck in postage, and you can sleep easy at night.
As far as music in the public domain - the music itself might be, but a given recording of it might not be. For example, music written by Mozart is in the public domain. The actual recording of "Eine Kleine Nachtmusik" performed by the ZenSteve Piano Company would not be; and won't be, for another 70+ years. However, you could record your own performance of the same public-domain music, and it would be yours.
Other options include hitting up the forums here (heck, there's even a composer who just posted right above this thread, on main page) looking for musicians trying to gather some work-experience. Or CraigsList.org. You can find all kinds of musicians willing to work for whatever, in exchange for building their resume. You'd still need to get a signed release from them, of course.
When you contact potential musicians/composers, just make sure you are upfront about money, if any. Get that out of the way first, or you'll ending wasting everyone's time (including your own).
