You have a scope. You could use it to to probe the speaker line. I don't expect to get beeps with no display of non-FF codes on the POST card.

The last 386SX board I revived had a broken trace causing a missing address bit on the ROM bios. Use your scope to probe the processor clock and ROM BIOS address and data pins. Watch out for strange waveforms (indicating floating inputs) and constant levels on signals that are unlikely to stay constant.

You need 4 pieces of SIMM-30 to work, 2 pcs won't work and neither would 1. For speaker, your POST card has one onboard, all you need is a wire to connect it up (which should come with your POST card). Btw, it horrifies me to see direct soldering onto the AT connector.

On a 386SX, you normally just need two 30-pin SIMMs, as your external databus is only 16-bit. However, I have seen 386SX boards that interleave the DRAM banks that do require 4.

Oops me bad, was thinking 486. Yeah 2 pcs SIMM would work for this board.

That soldering job with all the wires tangling left and right... brrr.
And what's with the strange wire colors? Red, orange and blue as GND?
What about beep codes? Get a speaker.
Get rid of that "PSU" and get a standard AT PSU first.

PSU is working fine. It's PSU originally with 24-pin ATX connector and voltages are stable. Colors has no matter but connections I made are compatible with AT pinout. Checked with multimeter. There are 4 GND pins, 4 +5V pins and others... If it will start I will mount standard AT connector. Now it's only for a try.

Ok. I will check something more with scope and let You know what's going on. I have inserted BIOS chip into programmer and there seems to be ok.

What is keyboard controller/bios chip there? I didn't reveal marking under sticker yet.

I guess that memory type can be a problem. There are recommended 9-chip modules instead of 3-chip.

One question: what are -5V and -12V lines for? Most of chips are +5V powered.

The "-5V" is for some ISA cards (such as the Soundblaster 2.0 CT1350B) that require it. Some mobos allegedly require it as well but not all of them do. Some DRAM chips also required it

The "-12V" was used for RS-232 (iirc), though I can't find reliable information as to what the negative 12V rail was used for.

Some Cards require -5/-12V for OpAmps for example.