How to test a power supply?

My computer powered off the other day on its own, and now when I push the power button, nothing happens. My assumption would naturally be that the power supply is done (possibly well done) but is there any good way to test this before I buy a new one?

Unplug the power supply from any of the components inside the computer (or just remove it from the computer completely).

USE CAUTION HERE (Though you'd only be shocked with a max of 24 volts)

1. Plug the power supply into the wall.
2. Find the big 24-ish pin connector that connects to the motherboard.
3. Connect the GREEN wire with the adjacent BLACK wire.
   (If you're colorblind, the green wire sits between three black wires on one end and one more on the other)
4. The power supply's fan should start up. If it doesn't then it's dead.
5. If the fan starts up, then it could be the motherboard that's dead. You can use a multimeter to check if there is power output from the power supply.

Most well-stocked geek-stores sell a "power-supply tester" that has all the appropiate connectors to plug each part of your PSU into, with spiffy LEDs indicating status of the various rails, connectors for IDE/SATA/floppy power cables, etc. They run ~$20 US.

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You could try using a multimeter to check the +/-5V and +/-12V rails. Due to the switching nature of the power supply, you'll want to put a resistor in series with your probes while measuring.

To test the power supply unit, follow these instructions from Corsair in the Power Supplies FAQ:

My Machine doesn’t start up when I hit the power button, is the PSU faulty?

Disconnect everything from your PSU except for one single fan that should be connected directly to one of the Molex connectors. Then, get a small piece of wire, paper clip, or suitable object and short the green pin and a black pin on the 24-pin connector on the power supply. The voltage present is a very low signaling voltage so no worries of being shocked. Your PSUs fan should spin along with the fan you have connected to it. If this is the case, your PSU may not be receiving the power on signal from your motherboard and you should consider other causes of the problem you’re having.

When a computer is turned on, the motherboard electronically connects the green pin (power on signal) to any one of the black connectors (ground). This signals the PSU to start supplying power. By shorting these two connectors, you are testing the PSU to determine if it can turn on when signaled to do so. If this doesn't work, then the PSU has probably failed. More information what each pin does on an ATX power supply unit is available on Wikipedia.

I have used one of the testers which showed a faulty supply to be good. The problem is the tester doesn't put a proper load on the PSU.

I replaced the supply and that fixed the problem. The voltage is OK if no load is applied. I serviced yachts for many years with 12 volt lighting. Corroded connections due to salt water are very common. Customers would return a new bulb and claim it was bad. I had to explain that 12 volts would show on their multimeter with the bulb removed, but this did not prove the bulb was bad. The corroded connection limited the current. Same principle.