I had this exact issue and had almost given up. I managed to fix it this evening, hopefully it works for you.
You can verify that we have the same issue by doing this:
1. Disable your Airport Express's wireless using AirPort Utility 5.6:
2. Connect it with Ethernet and verify that the connection is stable.
If this is the case, then the issue must specifically be with the WiFi and not something else going wrong inside the device. If you're still experiencing dropoffs in this configuration then sorry, I guess this won't help you.
The fix:
Update your router to use WPA2-PSK [AES]. This is the only configuration that worked for me. Second, set your AirPort Express to Join your network using WPA2 Personal security. Again, this is the only configuration that worked. The WPA/WPA2 Personal setting did not work.
Restart your router. It should work. My problem was identical to yours. I could see the AE in the AirPlay settings in iTunes for a few minutes after I booted it. Sometimes I could play to it, sometimes not. Once it stopped working, I would need to disconnect and reconnect the AE from power. The light would remain solid green.
But yeah, it works now. Good luck!