OK, so I've read the threads on the slim case 530s PSU failures, and I went through the recommended troubleshooting. I'm stumped, so here I am. You might get a kick out of this one. 
Background:
It was my brother's spare computer, mostly used by his daughters and if his wife needed to print something. He had it standing in a computer desk, and it was working fine. When he disconnected everything, pulled it out of the cubby, laid it flat, and reconnected everything (he got a new desk that had a different configuration), the computer would no longer POST. The light in the power button would turn solid orange and stay there, and the CPU fan would spin up for about a second, and there would be no other signs of life.
I had him blow out the case and try powering it up with various components unplugged. None of that changed the behavior. I had him bring the machine over for me to look at, where I basically did the same thing, only I unplugged everything right at the outset. No change. So, I pulled his HDD, rigged it into my Optiplex 745 (roughly the same vintage and form factor, but the mobo only has 2 SATA ports). He still had a Dell Vista install disk from when he bought the Inspiron, so I walked him through the data transfer and how to swap cables around to re-install Windows on the Inspiron HDD.
My steps:
I pulled everything out of the case except the mobo, removed all components from the mobo that didn't involve thermal paste (everything but the proc, heatsink, and fan), and plugged in a spare 20-pin ATX PSU that I have for testing. When the mobo turned on just fine (for having no RAM or boot drive), I re-inserted components one at a time, except for a HDD. Each time, the machine powered up. Bear in mind, this is with the original RAM that was in the machine. The only thing that didn't come with the machine is the CD-RW/DVDROM combo drive that I swapped from my Optiplex, so that my brother could keep the DVD burner.
I then took the screws out of the case of the PSU, took it outside, and blew it out. It was quite literally packed with dust and fluff! I reassembled it, disconnected the spare PSU, and reconnected the original, leaving it sitting across the removable structural rail. This time, all of the components were in place (again, minus a HDD), and, when I pressed the power button, the button turned orange for a split second before turning blue, and the fans all spun up for a two second count before everything shut down, including the green light on the back of the PSU. Then, it started back up again, going through the same steps, spun for two seconds, and shut down. Rinse and repeat. At no time does it seem to POST. None of this is with a monitor attached, because I don't have my proper workbench set up.
I let it go for a few times, but I didn't count. I was more surprised, first that it had started up, and then at what it was doing.
I thought that perhaps it wasn't grounding properly, not being installed into the case correctly. I reinstalled it, routed all the wire harnesses as they had originally been, and tried powering it on again, with the same result. This time, I removed components one at a time, each time getting the same cycle, until all components were removed. With no RAM, graphics card, or optical drive, it POSTed, but gave the two beep "Missing RAM" code.
I reinstalled everything except RAM, and I got the POST beeps, so I tried one stick of the two 1GB sticks in each slot, then the other stick in each, and this produced the cycling on and off. I then tried a spare, known good 1GB stick in each slot, again with the cycling each time.
So, I used to work for a Dell-authorized reseller, and I know my way around a soldering iron well enough, but it's been easily 15 years since I've troubleshot to the component level. I don't suppose anyone can point me at a schematic for a 24-pin ATX PSU. I can find pinouts 8 ways to Sunday, but I'm not even sure what the "extra" 4 pins are for, but I've found speculation regarding added power for the PCI-Express slot. I'm fine with pulling the mobo and throwing it into an old case with the 20-pin PSU and just setting it up with CentOS, but I feel like I'm tantalizingly close to fixing this and having a proper PC, again.