Tony,
From the sequence of photos, it appears that:
1. There are two frequencies coming from the unpowered setup: a low level signal of about 10 MHz (.1us/cycle) and a similar amplitude one of about 285kHz (3.5us/cycle) (photo1). This is RFI that must be coming from something else since all your circuits are powered off.
2. One of your circuits enhances the lower frequency noise with a larger glitch about every 3.5us (photos 2 and 3) - RFI coming from one of your circuits.
It seems likely that the lower frequency is related to the ESS operation but you'll have to study the clock frequencies. I can't figure out the ESS data sheet with regard to clock frequencies to save my life but maybe you can.
Last thought is that none of the interference or noise seems to be in the audio band or the result of a circuit malfunction. So, if you can't hear it and the system sounds great, then maybe this is what it's supposed to look like!
Kit
Ooops I goofed.
Was wondering where you got those frequencys from.
So first off the time base in all these shots is 0.05uS NOT 0.5uS
I see 100MHz and 7.7MHz, may be 100MHz and 15.4MHz.
This is RFI that must be coming from something else since all your circuits are powered off.
Maybe the scope has just had its day.
Shot 1 is also when the scope leads are just shorted, so that looks like its the scopes noise.
The burst noise that appearers when playing a stream I am happy with.
ESS clock is 100MHz which would have had me excited had it not been powered down!
LOts of other clocks in that lot, Arduino is connected on I2C, Rpi connected on I2S, LCD screen seems to have a clock and IS noisy, ESS internally up-samples, when running it had a 44.1kHz I2S input to 192kHz internally.
There are 14 regulators running (I switch from normal dual mono to straight stereo to simplify the procedure) the DUT is the quietest of the lot.
To my ears this single DAC setup sounds better than my previous dual mono but hoping "themysticals" golden ears will visit and confirm.
Taking shot 1 as a control measurement there does not (sadly) seem much point in proceeding further. As you say its all well beyond audio.