Check all your connections/cables first. Clicking noises are usually associated with a dying hard drive though. When PSU failure is intermittent like this, checking the voltages with a multimeter is pointless. So the next step would be to try a new PSU.
Since late September, my computer has been malfunctioning by not booting up when I press the power button. While my computer is connected, in the off position, I hear a repeated clicking noise. Occasionally, when I let the connected PC sit overnight while generating enough power, I can use my system as long as I do not turn it off. I suspect this is an issue with the power supply, but I have no idea how to trouble shoot this. This is a machine that I built, the specs are as followed:
CPU- AMD Athlon 64 3500+
MoBo- MSI K8N Neo4 Platinum/SLI
RAM- 1 GB DDR
Graphics Card- ATI x800 XL
Power Supply- AeroPower - The black / silver line 550 W
HDD- WD 74 GB Raptor 10,000 RPM