Some of the different issues you will see when a PSU fails are the PC does not turn on, HDD & fan do not spin, blue screen errors, overheating & random freezing, hanging or shutdowns. When troubleshooting a power supply, I would start by checking basic connections. Make sure the power supply switch is on, confirm that all cables are securely connected to both the power supply and the wall outlet. Then I would test the PSU form a different outlet to rule out the outlet as the problem. You could also grab your multimeter & test for 120V from the power outlet. If the computer still does not turn on, you can perform the paper clip test on pins 15 & 16 in the ATX connector while in the PC and see if the fan spins. If not, remove the PSU from the PC, keeping the paper clips in and then test individual pins for 3.3V, 5V, or 12V.

The other way you can test it is to buy a Power Supply tester from Micro Center and plug the appropriate 24 pin and 4 pin connectors in to, then plug it up to an outlet & turn it on. It will display the voltage coming of each pin. Either the light will be red for bad pins, or it will not light up at all, then you will know if the PSU is defective.