The Thumbplay Music streaming music service is now available for the iPhone, making it yet another cloud-based music service operating on the iPhone before Apple has yet to introduce its own.

Thumbplay Music launched in January in beta mode for the Blackberry, and has since expanded to Android platforms, and now iPhone. For $10 a month, users with the app can stream any song from its 9 million-track database on demand. It has an offline song caching feature for the BlackBerry, so users can play music while not connected to the mobile network, but has yet to bring that functionality to either the Android or iPhone versions.

The company has not yet revealed the number of downloads the combined apps have received, nor has it discussed overall subscriber numbers for the service.

The Thumbplay iPhone app joins similar on-demand streaming music iPhone apps from Spotify, Rhapsody and, among others, not to mention a bevy of webcasting apps from the likes of Spotify, Slacker and more.

iTunes has no streaming-based solution of its own. That may change when Steve Jobs makes his keynote June 7 for Apple’s Worldwide Developers Conference.