Music Maniac, you get to download free music for offline use.
With Spotify Premium you can download music and listen to it offline.
Theres also a android app you can get from their website, it's called YouMP34, it allows you to download music from YouTube. Just try not to download any copyrighted songs :)
Pretty much ANY app will. The difference is whether you will use you "Data" off you monthly or not.
I can listen to Spotify or Pandora on WiFi or just over my cellular data if there are enough bars of LTE. The android also comes with built in apps to listen to music you can move to your device, that require neither WiFi or cellular data.