The noise doesn't seem to be at any one frequency, but rather across the spectrum, with more low than high freq component. Scope and FFT shots below. Turning the wall-blister on and off doesn't seem to affect the noise - it seems to come entirely from the amp side of things.
I'm basically using the second circuit from here:
LM386 Schematic diagrams (the one with the 200 gain), with the bypass (pin 7) grounded directly, not through a cap.