Is it a tube amp? I'm wondering if one of your tubes is going microphonic? But, you say it doesn't happen when you're in a hum-cancelling position, which says it's RF interference. Those single coils are behaving like antennae and picking up the singer, another player with a wireless system or maybe even the in-ear monitors if you use those.
Seems to me there's a mod you can do to the amp to filter out RF without affecting the sound. Something like a .0005 cap to ground from the input. Maybe Mayfly remembers what I'm thinking of.