To cut a reasonably long story short, the Moog has prominent mains hum/whine-like spikes in its noise floor. They are there in main, sub and insert outputs, and are more pronounced when using an unbalanced lead - though are still there with a balanced connection. They fluctuate when adjusting hardware-related aspects of the synth, like LED or screen brightness, or changing cooling fan mode. They're audible if you crank monitoring levels, though at normal levels are sufficiently low in the noise floor that it's not a deal-breaker (even if it is a big irksome in principle - not a single other synth I own, analogue or digital, has anything like this).
Here's the real problem though: when I run the Moog through any external effects pedals (so inevitably using an unbalanced connection) the spikes get much more pronounced. For example, with a Strymon BigSky reverb, even though that is dead quiet by itself or when used with any other synths. Here's what I see on my MOTU 1248's analyser - absolute loudest signal peaks from the Moog in this setup go to about -36dB, average patches perhaps to around -50dB, so the S/R ratio is not good. Somehow those noise spikes are very insistent too:

Even weirder, some days ago I accidentally jiggled the synth's mains inlet plug (it's a lockable 4 pin XLR-like plug) while repatching some cables, and the spikes totally disappeared. I also discovered that they changed somewhat when simply touching the locking release button. Both of which suggest to me a ground loop issue of some kind. Via Source Distribution PMT sent me a replacement power brick, as the soldering in the original was pretty rough... But the new one does not improve the situation. With the same micro-jiggling it can still produce a spike/whine-free audio output (though never in the plug's fully inserted and locked position, so it's not practicable). This is the 'clean' analyser grab, that gives me a good 70-80dB S/N ratio even for quiet patches, and what I suspect the synth should be producing all the time:

In troubleshooting this I've tried multiple cables, unplugging all other mains in the studio, plugging the One, the effects, the 1248 and my mixer into different mains sockets, trying different IEC cables, comparing it with known good synths (Moog SUB37, Grandmother, VC340, Nord A1 - all quiet as mice), etc etc. Hours of headscratching. Only the One is misbehaving, so I can only really conclude it has a fundamental grounding issue - and then the question is whether it's a faulty unit or inherent in the design (which I could not live with, for my requirements).
Perhaps the only way to sort this out is to compare my synth with another One, and see if they behave differently.