For the scofflaws among us, there are endless places to illegally download whatever music you're looking for. But for those looking to expand their collections and musical horizons in more law-abiding ways, there are also lots of legit places to find music from new artists, from live concerts, and much more that won't cost you a dime. And that also won't get you sued by the RIAA in the process.
Online music stores
Apple and Amazon will gladly sell you any of its millions of songs or albums at the click of a button. But they also serve up a number of freebies as well. iTunes, for example, offers a free single of the week, featured prominently on the both the main and music sections of the iTunes Store. Apple has a page on the iTunes Store with all free iTunes content at any given moment, including music. And if you install the free Starbucks iOS app, you can download a free Pick of the Week (which of late hasn't exactly been weekly) directly from your mobile device (a different tune than you might get from visiting a Starbucks store). I've gotten tracks from Eric Clapton, Bruno Mars, Rod Stewart, and Norah Jones from it. And then of course there's the huge trove of music podcasts that provide daily or weekly downloads. (Note that all of the above options require a free iTunes Store account.)
Amazon MP3 actually has quite a bit of free content for the taking. There are pages for free songs, free albums (mostly samplers from indie labels), and music from rising stars'. As you might expect, much of it is stuff you've never heard of, but you'll find tracks from the likes of Blondie, She & Him, and The National, as well as classical tracks and kids' music.
Google Play offers a free music page--currently with 15 tracks from Busta Rhymes, Dave Matthews Band, The Rolling Stones, and a few other bands. The site also has a monthly Antenna sampler of music from up-and-comers, with 12 tracks for July, for example.
You can find free music in many, many other places. Artist websites, for example, often provide a free download or two to help you get to know them. But that's just the beginning.
Rolling Stone offers a free song every day of the week, mostly from lesser-known artists, but you'll find tracks by Femi Kuti and Skinny Puppy thrown in. AOL Music's Spinner Free MP3 Download of the Day showcases lots of unknowns. And NPR Music Downloads has some concerts and music podcasts you can download.
NoiseTrade provides free songs/EPs/albums/samplers from a wide cross-section of small/unknown artists. Each page has a "suggested tip" that you can opt to pay if you want to support the artists, and an email address is required to receive your download code.
Fans of South by Southwest (SXSW) can download a "mixtape" of 60 tracks by artists from the 2013 festival from Spin. And the Unofficial SXSW Torrents website includes more than 9000 songs (that's 45GB!) from 2005 to 2013 that were previously available on the SXSW site and collected for your downloading pleasure.
For fans of live music, now is a great time to be alive. Instead of snail-mail trades of cassettes or CDs, you can freely download more new and vintage recordings than you'll ever have time to listen to before you shuffle off this mortal coil.
The Internet Archive's Live Music Archive has, at last count, 118,340 items from 5740 bands--most of which you can download in MP3 and lossless FLAC formats (some stuff is streaming only). In the list of bands, you'll find music from such varied artists as The Grateful Dead, Smashing Pumpkins, John Mayer, Camper Van Beethoven, Little Feat, Maroon 5, Death Cab For Cutie, and Ween. Since bands must grant permission for their music to be there, LMA runs heavy on jam bands, but there's plenty to fill up the hard drive of any music lover. Most shows are audience recordings, although there are many soundboard recordings as well.
You'll also find a ton of live-music downloads on BitTorrent trackers such as bt.etree.org, Dime, and The Trader's Den. Of the three, bt.etree.org adheres pretty strictly to offering only material from bands that have a taping policy, such as Phish, moe., The Black Crowes, Primus, and Wilco. The other two sites include much more vintage material from every artist under the sun, and generally remove anything found to be on an official release or when a band complains. But if there's a band you like, there's a good chance you'll find a recording by visiting one of the above sites.
(And for more on live shows, check out "Collecting live music in the digital age".)