I had this exact same problem with my PC just a month ago. I cleaned out the fans, put in more powerful ones, removed some hardware... Everything you could Imagine. It still spontaneously shut down randomly, sometimes just leaving the Power LED on and a single fan (CPU fan) spinning. Here's how I fixed it:
I didn't have the money to go buy another 50 to 100 dollar PSU, so being the Electrician that I am I just opened the dam thing up.
Whenever a power converter, signal-inverter or PSU starts acting stupid, 99.100,000,000% of the time it's just a capacitor failing. You can tell when a capacitor has failed when it's either leaking, bulging up from the top, or caving in a lot. Here is an image comparing a good cap to a bad one:
The bad one's got to be replaced, and that's how I fixed mine. I opened it up, took a look around for a bulging capacitor, and sure enough, I found one. After I found out what Kind of capacitor I needed, I just went to radio shack and bought a new capacitor just like it, replaced the old one and problem solved.
If you do decide to go through the trouble of replacing the cap, you first need a few things:
- Rubber gloves, (a safety precaution to protect you from electric discharge, you never know..)
- Soldering iron (Never leave the thing unattended while it's hot or while it's cooling down; it could burn your house to a crisp..)
- Flux core solder (Flux is a paste that prevents oxidation, making it easier for the solder to bond to things..)
Also, please watch this video before doing anything. It gives you an Idea of what precautions to take, and more importantly what to look for.
Capacitor Replacement Tutorial - YouTube