There’s been Google, Google Images, Google Reader, Blog, Video, I can’t list all in the thread, but wouldn’t it be obvious for a sequel to Google Video:
Google Music?
What? I turned it off at the beggining. I got bored. 
It starts getting creepy after the things that have already happened
What would make a music sharing site as popular as YouTube without getting in illegal territory?
Though it would be funny to hear people who suck at singing, it may go to a site full of cussy music and immoral lerics that would force many people to condemn the site, note YouTube, I really feel like making a series of videos mentioning the videos I condemned that day to put on YouTube.
I have heard of people sticking there entire MP3 music library on youtube and using that as a Jukebox. each track typically last just 3.5 to 4 mins anyway and you can just send black for the vision.
Not exactly a “Google Music” site, but look at something like Pandora. (Its internet radio, which doesn’t let you search really, but eh…)
Legally, unless people are releasing their albums for free and/or under some sort of open distribution license Google couldn’t distribute music without paying for it. If you make a site where people can just upload their own recorded music, then that’s pretty much youtube or myspace. There’s really no reason to go into that niche when its already filled (and google owns one).
On top of that, who would really want to listen someone singing bad lyrics, with bad tone, etc., etc. ? Sure it could be funny, but once again, that’s why we have youtube.
So legally supply mainstream, copyrighted music - not likely. To provide independent artists with their own outlet, they already have.
Now if you’re talking about searching google for songs, there’s already ways to do that, but its illegal to download any of the results if they’re copyrighted (Is the act of searching for it illegal as well?). Don’t forget illegally downloaded files run (IIRC) at about $7.5 Million per 50 songs (~150 grand per song) in the U.S. thanks to some new legislation… (Pro Ip ACT of 2008: Here). In short do everything by the book and you’ll be good. Just don’t expect Google to buy it for you ;).
forTe: you might be interested in last.fm
you can search and listen and get recommendations but downloading is another story unless it’s listed as a free mp3