The music industry might be afraid of it, but there's a lot of free music to be found online. However, just because something can be downloaded doesn't mean that it's legal. There's plenty out there to listen to or download: free for the taking.
Peruse mp3 blogs. The mp3 blog aggregator Hype Machine (hype.non-standard.net) lists most every mp3 blog in existence with new ones being added all the time. Search for a band you like and you can play songs in a flash player.
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Use other aggregators, such as elbo.ws or del.icio.ous, to find mp3s. In del.icio.ous do a search with the tag: mp3.
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Play Internet radio. A simple Google search for Internet Radio will bring up hundreds of stations. The most popular are Last.fm and Live365. If you like Indie Rock, try Radio Indie Pop.
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Go to mp3 sites, such as mp3.com, Garageband, or Soundclick. Here's where you'll find music that's 100% legal to download. Record companies and bands upload sample songs so listeners can get a taste. It's possible to download music from an mp3 blog, but this is not necessarily on the up and up.
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Check out Pandora: a little different than the typical Internet radio station. You plug in a favorite band and the site plays similar music.
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Check out file sharing networks like Bittorrent. However, make sure the downloads are offered legally by the record label or band.
Tips & Warnings
You can hear clips of most any band on Amazon, Itunes, and the like. If you like a clip, do a search using the above steps to find the whole song.
Again, downloading music that is not free domain is illegal. Be wary of using services such as Bittorrent to download entire CDs.
Uploading music to which you don't have the copyright is also illegal.