Fixing an ATX PSU

Bad capacitors. It's always bad capacitors (and it's never Lupus). They've dried out and become resistive. In the process they sometimes make life ugly for power transistors and diodes.

If you're not safe with some mains voltage (there may be some in the hot side of the supply) please don't open the box.

Look for swollen electrolytic capacitors (metal cans). Replace with reputable parts with same values (no consumer components supplier will sell parts as nasty as a cheap manufacturer can buy wholesale).

Check diodes and transistors on output side for blowage.

Main high voltage mains side capacitor might be dried out. Shows up as overload in feedback paths and shutdown. Replace it. Must be same voltage rating!

Go see the sci.electronics.repair FAQ.

I'm going to go with Tim's answer. The caps are my first suspect. I once checked out a DVD player whose 5V line would pulse, much like your 12V line. Turned out to be a bad cap in the rectifier. I almost missed it, until I started touching the tops of all the caps with my finger (this is a good time to remind you to avoid getting yourself a Darwin award!). It was then immediately apparent which cap was bulging.