How can I diagnose a faulty power supply?

Wow, interesting question.

The problem with diagnosing a power supply problem (when it's not hilariously obvious), is that the problem will often only show up under load, so even if you pull it out and check the voltage on every plug, you may not see any problems.

As many others have said, you can swap it with a "Known Good" power supply and see if the problem persists. You might also want to try to run some sort of benchmarking application to put the system under heavy load: if the system always crashes under load, regardless of the application, that's a good indicator of power supply problems.

Assuming it's at least somewhat new (90's or newer) you can buy a tester Newegg has a good selection