Hey fellow frustrated airplay users, I've recently fixed this and wanted to share.
I'd never had this issue until our internet was moved from ADSL to NBN (Australia's fibre-optic network). When I installed the new service I also updated my Airport Express, so my first guess was that it was a recent update issue and they would fix it. Then after reading a bunch of replies to this thread it became clear the issue was with AE's automatic 2.4ghz and 5 ghz channels. Airplay fails to work properly on the 5 ghz network.
The fix is to split the two manually then use the 2.4 ghz network when you want to use airplay. To do this:
1. Open Airport Utility
2. Select your Airport Express and click Edit
3. Select the "Wireless" tab
4. Then click the "Wireless Options" button
5. Check/tick the box that says 5ghz network name. This then splits the two bands. I kept the name suggested by Apple, as this makes it simple but you can name the 5ghz as something else like "this is the wifi network that won't play music"
6. Click "save"
7. Click "update"
8. When your wifi reloads, select the network that's NOT named "{network name 5ghz}".
9. Enjoy music without the dropouts!!!