That is very bad. I recommend a comprehensive survey of the relevant parts of the electrical system, starting at your main panel.
Service grounding (earthing) system
This goes from a water pipe, ground rods, etc. to your main service panel. If you have a sub-panel, it goes to the main panel. It is the reference to zero voltage for the entire system. This goes to the ground in your panel.
Don't worry too much if it's not perfect, I'm more concerned with its relationship with the rest of your panel(s).
Ground-neutral bond in your main panel
Throughout the world, mains power is wired as if it's an isolated system with a grounded safety shield. That is to say, ground is never used in any way to carry current, except during a fault condition. But it is not an isolated system: to prevent wild voltages, neutral is intentionally bonded to ground in the main panel only, via a wire, strap or green screw in the neutral bus.
Make sure that exists and looks OK. Without it, a ground fault in equipment will "electrify your grounds". (actually put neutral at a wild voltage compared to ground.)
In the main panel only, it is legal to simply use one bus for all neutrals and grounds. It looks awful; it makes you think neutral and ground are redundant; they're not!
If these branch circuits are fed from a panel, which is itself fed from a breaker off the main panel, make sure the feeder has a separate white or gray neutral and green/bare ground wire. Metal conduit can substitute for a ground wire. Make sure the sub-panel has a neutral bus and a ground bus, carefully separated and not bonded.
Also in the panel
Watch out for a ground bus with wires that are not green or bare. Watch out for a neutral bus with wires that are not white or gray.
Now look for circuit breakers which have white, gray, green or bare wires going to them. White/gray to a breaker is OK under a few circumstances:
- a 2-pole breaker (the breaker itself is double normal width, and obviously has 2 handles tied together), often has 1 black and 1 white
- A GFCI/AFCI breaker, which has 1 hot and 2 neutrals going to it (one coiled up and going to the neutral bus).
Obviously, pay special attention to your difficult circuits.
Now follow the receptacle chain back to the panel
For each circuit in turn... Unplug everything and turn off one of the breakers. Use a nightlight, circuit tester, plain lamp and some intuition to identify which receptacles and other loads are on that circuit.
Physically check: Now, work forward from the panel to your problem location. Open up each junction box, remove the switches/receptacles from the boxes, and give it the basic eyeball. I bet you'll find problems at one of them.
Electrically check: Voltage testers are hokey and don't work, because they are only comparing the receptacle to itself. Noting that you started on electronics.se and know a thing or two, you can electrically test by running a wire back to that original zero reference. The zero reference won't harm you, but getting between it and a hot voltage could, so I'd add a 20k-50k ohm resistor on the far end of it.
- Voltage to ground should be <0.1 volts, essentially 0.
- Voltage to the taller neutral slot should be 0 to 3 volts depending on load on the circuit
- Voltage to the shorter "hot" slot should be near 120 volts
- If you add a ~1500W heavy load on the circuit, neutral should increase, and ground shouldn't.
This is no substitute for a physical inspection. You can't reliably detect a bootleg ground or ground-neutral swap, for instance, which are things which would cause this.
I guarantee you'll find a problem before you're done. Possibly 2 or more, since a guy who makes a mistake often repeats it.