Computer Trouble (strange problem)

[Larson2042]

Recently, a household computer was having troubles which were fully resolved by replacing the power supply, which had apparently gone bad. It was replaced with a 480W Antec. Everything was fine until tonight when the screen went black, and a screeching noise started in the speakers. So, I reached over to turn the power supply off, and lo and behold, switching the power supply off does nothing. After pulling the power cord and then replacing it and experimenting, it does not matter if the switch on the power supply is on or off, the computer will still turn on. This confuses me greatly, since I would have thought a switch like that would either complete a circuit or not, or similar sort of thing.

And now, though the computer does turn on, it does not boot. It doesn't even get as far as the bios posting. All the fans turns on and such, but that's about it. The screen remains black. (using a different power supply does not change this, but at least its switch actually turns the computer off)

So, I'd like to know what you all think. Is it just a case of another power supply going bad randomly, and I'm just unlucky? Or would, say, an iffy motherboard be able to cause something to go wrong in the power supply like this? And I'm also sure that there's either a motherboard or processor bad now, as well.

Computer specs:
2.8 Ghz P4 (at stock speeds)
1 gig ram (cheap stuff, but I know it is still good)
Radeon 9800Pro
MSI 875P Neo Motherboard