Bah!

The scratch plate has just come loose on the Gretsch. I tried to remove the whole thing to get at the screws but I am not in the mood.

Smeg from King Kurt has a Gretsch, a proper vintage version of my green Single Anniversary, and he fixes his with Blu-Tack. If its good enough for him, its good enough for me. So that's exactly what I've done, and it works fine.