Google Play Music: At first glance, this seems like a subscription service and somewhere to buy music. I skip all that and use Google Play as a back-up for my personal music collection. When you click on the "My Music" option, you can store up to 50,000 songs and listen to them from the web anywhere. The awesome advantages are that I have a complete back-up of my music collection and I can listen to it from any device, anywhere! I can make playlists, too. If you own CDs and/or MP3s, this app is worth a look.
Songza: This is a music streaming and recommendation service offering curated playlists based on the time of day or a user's mood or activity. The way I use it is to choose a category and play that. There is no limit as to how much music you can play and there are apps for all devices or you can listen from a browser. It's a pretty sweet music app for free music.
Spotify: Stream just about any type of music for free with Spotify. There are quick ads about every half hour. This is seriously one of the most easiest ways to grab just about any song you can think of, and great for discovering new music. You can make playlists or play categories. There are apps available for all devices or listen from a browser.
I have listed my top choices, but there are many more music apps available. You might want to try one of these:
Slacker Radio
Last.fm
Pandora
