Why does my shower handle give me a little zap when my fingernail touches it?

Either way, don't sit on it. Mild today doesn't stay mild if a wire keeps rubbing through its insulation.

Had almost the same thing with a kitchen faucet years ago, tingle only when my fingers were damp or the contact point was small. Turned out to be a dishwasher sharing a neutral that had gone dodgy, and the pipes were carrying the return. Small contact points feel it more than a fat fingertip, that part tracks. The thing that solved it was an electrician with a clamp meter, found it in ten minutes. Also to your GFCI question, they only trip past a certain threshold, a slow drip of leakage can ride right under it for years. Worth having someone actually measure it instead of guessing.

That nail thing makes sense, the edge of a nail concentrates the current into a tiny spot so even a couple volts feels like a zap. When I measured one of these on a job I read 4 volts AC between the trim and the drain, enough to feel, not enough to trip anything. Grab a cheap multimeter and check trim to a grounded tap with the water running. Also, a 70s house can easily have the shower light and fan on a circuit with no GFCI at all, so the fact yours never popped tells you almost nothing.

Still figuring out how forums work here so bear with me. We had somethig similar at my place but it was the laundry sink tap, only buzzed when the washer was running. Turned out the old copper water line had been replaced with PEX partway and the bonding got left behind, so the pipes were just floating. An electrician re-bonded it and the tingle went away. Since yours is a 70s house, is it worth checking wheather the metal plumbing is acutally bonded anymore? I don't know if that woud explain your shower or if it's more likely something live touching the trim, but maybe someone here knows if a missing bond alone can cause that.

Not sure I buy the 'not enugh to trip anything' line from the multimeter reply. A GFCI doesn't trip on voltage between the trim and the drain, it trips on current leaking somewhere it shouldn't, and the trip level is way below what you'd feel as a tingle. So the OP's outlet never popping tells you almost nothing. I had a hot tub light doing the same low-buzz thing for months on a GFCI-protected circuit and it never once tripped til the element actually shorted. The meter check is still the right move, just don't read a quiet GFCI as all clear.