I have a Transformer Prime running Ice Cream Sandwich. When I start up Spotify, it goes online for a brief while (0.1sec?) and then the offline bar at the bottom of the screen appears, then the "offline bar" disappears, and returns, and goes away, and returns. Eventually the app stays offline.
This online-offline-online-offline behaviour was the case with the official app. Then I upgraded to the preview, and spotify still exhibits this behaviour.
I also have a Samsung Galaxy S with the official client, and it stays online and plays offline playlists, streams, etc etc.. So does my logitech squeezebox. And the Spotify application as well. (Yes, I have a bit of a gadget obsession).