Spotify's Android app lets premium users listen to music on their phones, both online and off. Photo courtesy of Spotify

Previously available only in Europe, the music service launched Thursday in the United States to much applause. On the service’s desktop client, you’re able to listen to any of up to 15 million hosted tracks, on demand, absolutely free.

But the real gem isn’t found on the desktop app: It’s on your phone.

Spotify’s U.S. release arrives with both Android and iOS apps, which allow you to stream music wirelessly to your s...