Now that I know what it looks like it's easy to spot. A good capacitor has a nice, perfectly flat top; a blown capacitor has a top which pokes up a bit. Blown capacitor = Dead Motherboard.
In a nutshell: How to determine why your computer crashing
This is the process I went through to figure out what was wrong with my system. Luckily I previously downloaded a few tools to help:
1) The easiest thing was to run a memory-checker from a bootable CD. That ran for an hour of memory checking before I decided memory probably wasn't the problem. But as time went on the system was crashing more and more (this is generally the sign of a bad motherboard or power supply).
2) Is it the operating system? On my system I booted up Knoppix Live (Linux) via bootable CD. Within a few minutes the computer crashed. Guess Windows was not to blame.
3) Before booting Knoppix I had disconnected the hard drive (with system powered off of course). But even with Knoppix running of CD the problem was still there so I knew it was not hard-drive related.
4) My motherboard had on-board video, so I also tried removing my video card. It didn't help.
5) So now it must be power-supply or motherboard. Power-supply is more likely and easier to swap out for a test.
At at this point I took mine to the local computer store (they had offered to swap out the power and see if it worked). He took one look at the motherboard, saw the capacitor blown and knew it was the problem.
