PC possessed by ghosts (or psu problem)

The last few days my pc has woken me up in the night by turning it's self on. It powers up, gets quiet, then powers down, and it just keeps doing it. I usually just pull the plug on the psu and it works fine in the morning. Now, however, the power supply turns on whenever plugged in (nothing else in the pc turns on though, save for the lights in the front and on the motherboard.
Is this a major problem? How would I go about fixing it?

I do occasional work for a school where their whole lab is "haunted". You'll just be sitting there and one will turn on, then randomly other ones in the room will do the same. I'm certain it's not a wake-on-LAN thing (as it's turned off in BIOS) - I reckon the vibration of people walking around, or even the vibration of another machine turning on, fans and hard disk starting, etc., or maybe even the degauss coil in the monitors, is closing poorly-gapped switches and turning the things on.