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I'm going to try borrowing a PSU from somewhere to see if my problem lies there. However, all kinds of helpful suggestions are welcome.
Your hunches are exactly on the right track. Problems with unreliability turning on leading to eventual total failure to power up are classic signs of a failed PSU. PSUs are actually the most common part to fail in desktop computer systems and during my time working as a computer technician in a school, that was definitely the part I changed most often (followed some way behind with hard disk drives). Fortunately you have a well known brand but they can be a great pain if they're a poor design and if they fail spectacularly because in that sort of situation they sometimes end up taking other parts with them. Modern, well designed PSUs have all sorts of protection measures built in to help minimise the chances of this happening though.
Make sure the replacement PSU has adequate power for your system and you should be up and running again. You mentioned you noticed the extension light was flickering - if you are having problems with an irregular power supply, you should definitely consider buying an Uninteruptable Power Supply (UPS) which will keep your PC running in the event of power cuts or brown outs (at least for a few minutes, long enough to safely shut it down) A UPS will also help reduce the strain on the PSU during normal operation by providing a smooth stable power supply to the PSU at all times. If you want to be really safe, you could also buy a surge suppressor and put that before the UPS and that will help protect you if there are lightning strikes or other situations that cause the mains to run in an over voltage situation.