PS2:Power Supply Repair

Original/Phat PS2s
If the PS2 doesn't turn on (no LEDs/no fan sound/nothing), then there is almost certainly a problem somewhere between the power cord, power switch, and power supply board, which should output 12V to the main board. Given the age of the PS2, the most likely issue is failing electronic capacitors, but we shall see. By the standards of switching power supplies, the unit in PS2 is simple and doesn't have many components.

The original/phat PS2s feature a PSU board that can be replaced, and replacements are available. But given how easy it might be to fix, perhaps you can avoid contributing to e-waste and hopefully just replace a capacitor or two. Note, however, that if you're probing around in live power supply in any kind of consumer electronics, you are risking potentially lethal electrocution. If that risk isn't obvious to you, you'll want the help of a friend or professional who is comfortable with mains voltage (115V and up). Alternatively, you can simply unplug the PSU, replace components, and then plug it in without probing, which is safe, but you'll have less diagnostic information.

With the PS2 unplugged and dissembled, inspect the obvious things:
 * 1) Are there glaringly signs of catastrophe? (blackened components)
 * 2) Does the power cord work? (try a known working cord, or for continuity)
 * 3) Does the power switch work? (test for continuity, replace if faulty)
 * 4) Is the fuse still intact? (test for continuity)
 * 5) Are electronic capacitors bulging or covered with a crystalline fuzz that suggests leakage?

If the answer to all the above is no, the most likely issue is that capacitor C20 and maybe C3 have gone bad (often they go bad without any visual signs). Replacing them is extremely basic soldering 101 stuff, and it can't hurt. Get a new capacitor with the same capacitance rating, and a voltage rating equal or higher to the bad cap. If you have caps with sufficient voltage ratings, but not enough capacitance, you could put 2-4 in parallel to sum up to the required capacitance. Do not use a cap with a lower voltage rating! Although measuring leaky capacitors is not something a normal multimeter can do, one of those cheap LCR-T4 meters will probably reveal the cap to be way out of spec.

Other Scenarios
 * If there's blackened components, you'll need to understand the original thing that failed and the cascade of damage it caused, replacing everything along the way.
 * If the fuse blew, it blew for a reason, i.e. something else is wrong
 * Diodes can be checked with a meter. If they give fishy readings, desolder one lead and test again to confirm
 * The mosfets could have failed

If you're interested in live testing a power supply, watch videos from professionals (eg. Mr. Carlson's lab on YouTube, who covers both the safety protocols and the theory of how switching power supplies work).