I had this problem and I believe the problem has to do with the Video and/or the NetStream classes in the Air SDK.
Once you create a Video class like new Video(), and later bring up the in-app dialog, or any dialog that sits on top of your app, the app display goes black. The solution at the current time is to not use these classes. Adobe fixed this for some devices in air 3.5 by adding <containsVideo>true</containsVideo> to your <project name>-app-.xml. The fix works for some devices but breaks other devices so bad that the black screen problem happens regardless of Video/NetStream usage.
Right now the only solution I know is to not use the Video/Netstream classes (was using them to play FLV video files) and to be safe, use <containsVideo>false</containsVideo>.
Here is a related bug report: https://bugbase.adobe.com/index.cfm?event=bug&id=3703836
Note this issue has nothing to due with Starling as far as I can tell.
If you run the Android SDK command (not to be confused with the adobe Air command): "adt logcat" look for V/VideoSurfaceView(): surfaceDestroyed
If this happens, your app likely has gone black.