Strange problem

[The Magic Pudding]

I don't pretend to know what's wrong, motherboard or power supply maybe.
There is a technique for checking power supplies if you've got a multi metre.
Easy things to do would be to check the cord from power point to case, try another cord I've had powerboards with one dodgy socket, you could try plugging it into another socket.
The internal power connection from power supply to motherboard could be checked.
Power supplies often have a switch, I've never seen a wobbly one but I haven't seen many things.

I don't use a UPS, I don't know if they could do this, probably not, don't know.
I haven't got one of those fancy power boards that turn themselves off either.

I suspect some annoying motherboard problem that may get worse.
Video card, hard disk and memory problems shouldn't stop the computer making some noise.