Why won't this PC power supply run my BLDC?
I figured out what the problem was.

CLUES


There would be a clicking noise when I was using the 12V output and it would run for a moment before shutting down. 


This hinted that the power draw was not constant.

My BLDC would run when connected to the 5V output and there were no clicking noises.


This hinted that the problem was to do with the amount of current being drawn.

The power supply would stay off until I hit the switch off and then back on.


This hinted that it was due to a safety feature since it entered a mode where functionality was ceased completely until I restarted the supply.



Solution

One of these:


A large capacitor connected from the positive to the negative feed of the power supply smoothed out the sudden changes in current draw and fixed the problem.

Now there is no clicking noise and the motor runs just perfectly.

I hope this helped someone else.