Well as far as short circuits go, I'd look for anything which looks like it might be capable of conducting electricity, hanging out in a place where it doesn't belong. Also, maybe look any wires which might of worked their way out of a plug, a fan that appears to have stopped working, and or anything which has a burnt smell to it.
Also, they sell inexpensive power supply testers on Amazon, and they can take the guess work out of trying to figure out if the power supply has actually failed, or if it is something else. Just curious, do you live in a country that uses 220 line voltage?
If you remove the power supply from inside your computer, then open it's case, it will be obvious if you happen to have any capacitors which are going bad, or have already failed. First the capacitors start to bulge, and then later on they actually split open and leak their innards inside the power supply.
Well, you can unplug the power switch from your motherboard, and then you you take a small metal object and touch it to both of the motherboard pins for the power switch. If the computer starts, then it was the power switch causing your trouble. If you decide to try this test, make sure you don't touch your metal tool onto anything else than just the two mother board pins for the power switch.
Hope this helps.