In my limited experience-
MOBO's failing usually start with a device failure. Something that is there and has been working suddenly appears not to be there. (IDE controller, sound or video card etc)
Device manager will show problems.
Sometimes looking in the BIOS can tell you what's wrong. (again, checking the BIOS for power supply output can give you a heads up on power supplies going bad too).

Memory can be trickier without testing it. Systems will try to run even when mem chips start going bad, just lots of errors occurring or performance suddenly lacking.

Processors? Never had one go, ever. They seem to outlast most other components, unless overclocked, underfanned or burned by the power supply.

For many device failures, the power supply should still be considered as an early option.
Hard drive acting funny? Check voltage to it. Drive may be fine, just not getting correct power. I've had more than one drive I thought went bad come back to life with a new power supply.

Make your sound your own!
.. I do not work here, but the benefits are still awesome