Now that I have my computer at home (yes, I am posting in a negligee, with ostrich-feather trim) with a CD player and all, I’m interested in finding (legal!) sites where I can play and/or download music–particularly, pop songs of the 1950s and '60s (I already have bookmarked the wonderful tinfoil.com, where I can listen to music from the 1890s-1910s).
Most of the songs you know from the 50s and 60s are still under copyright and thus there is no legal way to download them. You can try http://www.mp3.com to get free music, however.
(It’s very interesting that people who support file sharing as a way to support up-and-coming artists, aren’t interested in downloading from MP3.com, which does support new artists.)
I think it would be pretty hard to find legal sites for '50’s/'60’s songs as they are copyrighted, as RealityChuck said. Another site for up and coming artists www.garageband.com
You can try the free downloads section of Amazon.com too. Most of the free songs they have are newer, but if you hunt a little you can find some older Jazz, Broadway, etc.
There is also allofmp3.com . It is approximately the same price as emusic, but allows you to download in any bitrate desired and also has a great selection. I have not tried it myself, but it looks like a good deal.