Can be anything from connectors or cards not being properly seated to defective components.
My first suspect would be the powersupply, if you have a multimeter you can check if you get good voltages on the motherboard.

Okay. A little more info. It's a P3B-F v1.03. The green light comes on on the motherboard. If I plug in a hard disc, it spins up and the head actuator moves.

The CPU is a 1400 MHz Tualatin in an Upgradeware adapter. That's had the paste cleaned off and replaced.

This used to be a functioning computer, some time ago, but has been idle for a while. The power supply blew when trying to start it, so that's already been replaced.

Tried a Geforce 4 MX too. No good.

Tried a subset of the RAM, but no change.

If I try it without the CPU, I get a high/low alternating beep. I've had no other POST codes.

Is there anything I can check to pin things down further?

I'd probably try another slot 1 CPU if you are able to get your hands on one, to rule out any weirdness with the slotket.

also grabbing a PCI / ISA diagnostic card would probably help a lot. I got one and it has served me well

EDIT: mmh..P3B-F... PSU blown when you powered it up...could it have done any damage?

The symptoms are not exactly the same but maybe give a look at this post and mine 2 below that one. I've had issues with a P3B-F and both another user and myself found a similar blown trace. Maybe worth inspecting yours?

P3B-F stopped working...need help

EDIT #2: also look at varrol's messages at the very bottom of the thread. Same symptoms as you. No fix 🙁 but maybe worth a PM if he progressed since then without updating his posts?

EDIT #3: 😁 I'd also suggest a mod moves this to Marvin\General Old Hardware, probably more likely to get knowledgeable people to see it compared to here.

I own 2 slotkets. The first slotket that I bought did not work on my motherboard with a pin-modded 1.4 GHz Tualatin Celeron, but it worked fine on my motherboard with a 1.1 GHz Coppermine-128 Celeron. Then, I bought a different kind of slotket, and it works fine on my motherboard with the pin-modded Tualatin Celeron.

TL;DR
Some slotkets work with Tualatin CPUs and others do not. Also, you may need to pin-mod your CPU.