Where are you getting that three beeps means memory failure?
Does it seem thermally related i.e. does turning it off and letting it cool for a while bring it back to operation?

What do you mean 66i/50i VLB?

Any corrosion on any part of the board?

I had the same problem with Soyo 019R1 386 motherboard. It was fine before shipping and the first boot gave me POST, but then it started to dergade. At one point it was giving 3 short signals only after the CMOS was cleared. Tried various VGA cards, different cache, different RAM, 50MHz OSC (instead of stock 80MHz), all jumper settings that were possible - nothing helped. It ended when the board refused to POST under any circumstances. I stripped it clean and threw away... It was a good board, a shame it ended up like that.

66i/50i means 33x2 and 25x2, "i" is for internal, in other words, CPU frequency.

I'm basing the beep code off of standard AMIBIOS diagnostic codes, which is part of the reason I'm confused, since those error codes are beeped out in a single burst and in a lower tone... the closest thing I could find is that it represents a base 64K memory error, but I doubt that's accurate.

No corrosion that I can see; the board seems to be in physically pristine condition. The clock battery has probably been thoroughly dead for the past ten years, but I doubt that would cause a total system failure on this level. I'll stick a picture of the whole thing up here when I get a chance.

Yep, I know VLB cards tend to be a bit sensitive at higher clock speeds; this 486DX2 runs at 66Mhz at an x2 multiplier, giving it a 33Mhz internal clock speed... dropping that down to 25Mhz didn't help any.

The video card is definitely on its way out, though... I might try running it under the hot air reflow gun for a while, but even if that does work, it's a temporary fix at best. I'm going to give it one more try with one of the most basic ISA video cards I can get my hands on - probably a Western Digital Paradise 512K - that I know is definitely functional. I have 1MB (8 128Kx8+TAGRAM) of cache on its way too, so I'll be able to check that off the list once it gets here.

On the long run, I would suggest keeping a few spare parts of every major component. High compatibility and good working condition over high quality and speed, so you do not have to spend a lot on the spares. This is the only way to ensure a more-or-less conclusive troubleshooting on your own.

In your shoes,
1. I would first check the CPU and RAM, (maybe cache chips as well) in an etalon-ly good motherboard
2. clear CMOS, recheck jumper settings, upgrade BIOS if necessary/possible on the subject motherboard
2. then put a simple ISA VGA and ISA multi I/O card in the subject motherboard
3. try POST and boot from a floppy or IDE HDD
4. run the usual diags / benches

when everything works, you can start adding stuff *one-by-one* like SCSI devices and permutate their resources, changing slots, jumpers etc...

First try with the minimum configuration and go from there.

Minimum is mobo+cpu+ram+vga. Memory and cache with slowest timings. Simple ISA VGA card. Try different ISA slots to get stable operation. Only then you try faster memory/cache timings and see if stability changes. Then you add I/O controller and boot from FDD/HDD. Try different benchmarks to test stability (3dbench, pcplayer, quake...). After that you change to VLB cards... and so on.

486's are temperamental. Like everyone else I'd try the minimum CPU+Ram+Isa Video. Even try different combinations of your RAM.
Hopefully It'll become stable enough to start trying I/O controllers. That said my 486 does funny things without any HDD plugged in, even if its not actually setup it bios