I don't believe I said anything about checking for shorts.
I said a certain behavior could indicate either a defective
supply, an overloaded supply, or a shorted voltage. We need
details of that behavior before we can move to the next step.

If you believe you have a short, then first you can say
which voltage is shorted. Again, that is what the meter would
have shown. The one voltage and did not rise is the suspect
voltage that would also cause other voltages to be shut off.
Again, which voltage? Given that information, then we can
continue to the next procedure.

The multimeter is not likely to be very good at detecting
shorts - depending on its design. Better meters will check
continuity with using less than 0.7 volts and less than 5
milliamps. No continuity meter better output the full nine
volts or even 3 volts. To the trivial current output by a
meter in conductivity mode, the entire computer is likely to
consume all that and more - therefore look just like a short.

Once we have information from previous readings, then we can
determine (without removing any power supply) if power supply
is overloaded, failing, or shorted.

Again, how to increase labor significantly. Remove the
power supply to test on a bench. Literally anything a bench
test is going to find - and more - is done with the supply
left in the computer.

Al wrote:
Thanks kindly for all of the replies to my previous question. I've
read twice through each reply - some will take another run-through
tomorrow to really absorb, since I do want to understand it all
completely

But, I just did an experiment which tells me that the PS is probably
not the problem. I borrowed a Compaq and pulled its working ATX PS.
The resulting behavior was the same - power on for a moment only.

Does this pretty much say that the PS is not the problem here? What
should be my next step in diagnosis?

From the reply by w_tom1, it seems I must check all the pins. Tom
says to check for shorts - but I have only a 9v analog powered
multimeter, so I'd think I can't use the ohm meter, except on 12v
circuits, yes? Or do I use the voltmeter and look for what pins
stay mostly flat on voltage and which pins might spike? Or does
the analog meter (an old Radio Shack one) respond too slowly to
tell me anything?

Thanks again for all replies.