At $9.99 a month for a premium subscription (necessary for using on mobile devices, but free with ads or $4.99/month without ads otherwise), Spotify not only lets you create your own playlists of favorites (or "starred" songs) and full albums, but browse through strangers' playlists that are public as well as your Facebook friends' playlists (should they choose to publish them). This one also shows you what's new every week, and can create a Pandora-like radio experience, playing similar artists' songs together. You need the premium one to get to all of your tunes on mobile devices, but if you're okay with just radio, you can pay the $4.99 a month. We'd recommend the whole package, though.
Rdio is similar to Spotify, but to listen to the music at all, you have to fork over $4.99 a month. That gives you unlimited web streaming on your browser/laptop/desktop. For $9.99 a month, you get web streaming as well as mobile streaming, so it's the same cost as Spotify for the mobile service, but Rdio doesn't offer a free plan at all. They do, however, offer a "family" deal of $17.99 a month, which allows membership for two people, or $22.99 for three people—a pretty good deal. Between Rdio and Spotify, there's not much difference at the $9.99 level, but otherwise there are a few different choices.
With Rhapsody, you can listen not only on your computer or on your phone/tablet, but also in your car or on your television if you have the proper connectivity cables. It's the cheapest when it comes to subscriptions—$9.99 like the rest for unlimited music streaming and no ads, but for $14.99 a month, you can download/listen to all the music you want on three devices, not just one. Also, with Rhapsody, your first 14 days are free and you aren't charged until after the two-week trial period is up.
Grooveshark is probably the least efficient of all these apps, since you have to manually create playlists of everything, even if you just want to listen to one artist's new record all the way through. Dragging-and-dropping individual songs is sort of a pain. That said, this one has music that the other three don't sometimes, so it's nice to find stuff you wouldn't otherwise have access to. Still, though, the other three win out. Grooveshark is free online, but for $6 you can eliminate ads, and for $9 a month you can bring it with you on your phone.