I got the Corsair RM850X, installed it, and took photos to make sure I knew exactly what was plugged in where. Booted successfully, ran some checks on Windows and then started the Rift and loaded Air car. It seemed to work a treat, but then....
the computer shut itself down again. With the old PSU, I used to be able to boot back in, no problems. This time, however, it wouldn't turn back on. I turned off the power switch, waited a few seconds, then tried again. It turned on, but then I noticed traces of smoke come out of the top and I turned it off immediately...
Oh crap, what have I done?? Any suggestions?
You let the magic smoke out! It won't work when it escapes.
When you have strange random crashes and BSODs, it's often very difficult to diagnose. People do overlook the PSU when diagnosing, but it's notoriously hard to pin down PSU issues if it's just not quite delivering enough current, or the voltage varies at random times, or it otherwise provides a "dirty" power signal.
But those sorts of issues can also be caused by overheating, unreliable memory, or a failing motherboard. The latter, the motherboard, can also result in sudden shutdowns. These days the MB acts as a direct "power switch" to the PSU, via shorting the green pin on the large motherboard power connector. If it disconnects that pin for any reason, the PSU will shut off.
If you saw smoke coming from the top, we can't 100% rule out the new PSU, though that would be a vary rare situation (DOA to the point of burning out instantly). I'm suspecting your motherboard now.
So the first steps should be to open the case back up, and look for signs of something burned. Bad capacitors are a common problem on motherboards. But there could be dust or dirt that shorted somewhere, or a loose screw that was sitting on something it shouldn't, or something. If the motherboard burnt out on you, you might still have OK components (CPU and RAM), but it's also possible that it took those out with it.
If you have the courage to power on again, take note of whether all of the fans are still turning, including the CPU fan.