I just got a Traynor YBA-1A, and it has a serious hum issue.

When you power on the amp (even unplugged from speakers) you can hear the transformers humming. This isn't intrusive, but it is the same noise the comes out of the speaker, so I suspect they are related.

Plugging in the speakers, you hear a hum even with the standby on.

Flip the standby switch, and it is loud. Quite loud. Loud enough to make the amp rather useless below about 3 on the volume knob. It isn't in the instrument path, even with the dials all at 0 and nothing plugged in, it hums just as loudly as it does halfway up the volume dial with a (single coil!) bass plugged in.

Pulling V1 and V2 preamp tubes does nothing to quiet the hum. Pulling V3 (the phase inverter, I assume) quiets it down somewhat, but it is still quite loud.

Swapping the power tubes changes the tone of the hum, but doesn't decrease it. With the Chinese tubes in it when I bought it, turning the treble dial to the top actually quieted the hum significantly. None of the other dials affect it at all. But this dime the treble trick doesn't work with the JJs I have in there now. (I'm talking power tubes, of course. Preamps have no effect on this)

I've tried desoldering the fan because it is AC, this does nothing.

I'm sure it could benefit from some new caps, as the ones in here are still originals (the amp is dated 1974) but that would more cause crackle and ghost notes, not an AC hum, I think... right?

What should I try next?
I'm not an amp tech but it seems to me that you have a grounding issue. Unless you are an amp tech, I'd take it to one and have the circuit traced out to check all grounds. Does the amp have a 2 plug power cable? Have you tried it in different outlets? It could very well need new capacitors. Hopefully it does not need new transformers. Have you spoken to the previous owner? Congrats none the less.
I will take a look around the circuits for that. The ground pin is snipped from the power cable, so that was one of the first things I tried (basically, jumping a wire from the shell of this amp to the shell of another amp plugged in) which produced some noise then the contact was made, but didn't kill the hum. I'm still going to replace the plug, but I doubt it will do much for the hum.

The previous owner mentioned the hum, so it isn't just doing it here.

I'm worried about the transformers. As I mentioned before, they both hum audibly. Google searches don't seem to give me much consensus on if audible hums are related to electric hums. The audible hum isn't loud enough to be a problem, unless they are related.
I'd recommend a capacitor change. It's a '74? Those caps have well over 30 years on them. The electrolyic in those caps is dried out, which is why you're hearing the hum - they're just not performing any more. Take it in for a cap change and it should be all better.
Well, I am going to change the caps, because they aren't too pricey, and it does get some ghost notes on high guitar notes. We'll see if that clears it up. I was under the impression that replacing caps won't cure hum though.
Does it have a 3 prong plug or the old 2 prong? Taking it to a tech and having the caps swapped and proper grounding put in would help regardless.
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It is a 3 prong, although someone snipped the ground pin. I've tried a grounding wire between the chassis and pin 3. That doesn't help much.