If you want to go further in diagnostics, grab a multimeter (absolutely any multimeter will do) and probe the gray pin in the 24-pin connector and ground. This pin is PWR_OK (or POWER_GOOD) signal coming from the PSU, it should be 5v. I've seen Power Supplies thrown away because people thought they were bad, and it might actually be something like a loose power connector from the wall, or even the voltage selector switch, if said PSU has one.
This test may not apply if you buy a really cheap PSU.
-Edit: To check if the PSU has a proper self-test circuitry, you can backprobe for continuity in both the gray pin and any 5v rail. There should be at least some significant ressistance. If the resistance is near-zero, the PSU doesn't have any sort of Self-test and should stay away from it...
Aidymalc reacted to TheXDS in README: How to respond to a no POST or no power up situation