Diagnose issues with your Windows PC hardware
A short list of free diagnostic tools.
Thanks to Doug (who wishes to remain mostly anonymous) for his software suggestions, talking me through the initial diagnostic process, and for reviewing this article.
Note: this post links a bunch of software and tests you can run. Some of them stress your system in various ways. I am not an expert at this stuff. Use at your own risk.
My desktop recently became reluctant to switch on. Once it was on it appeared to be fine, but clearly something wasn't right. Unfortunately, that something is probably a loose wire that'll be hard to access and fix (even if I was prepared to - soldering is where I tend to quit). But while working through trying to diagnose the issue, the exceedingly helpful Doug introduced me to a bunch of useful free diagnostic tools for Windows PCs. I've listed these tools here for future reference.
GPU (and PSU)
FurMark stress tests your GPU to benchmark it. This has the helpful side-effect of drawing a lot of power, and is likely to expose any issues with your PSU.
RAM
Windows Memory Diagnostic is pre-installed on Windows 10. It scans your RAM to check for issues. It requires a reboot.
SSDs and HDDs
CrystalDiskMark tests the performance of your hard drives. This can help you see if they're failing.
Voltages, temperatures, fans
HWMonitor reports on your computer's voltages, temperatures, and fan speeds.