I think it's switching noise from a DC-DC converter mainly. No idea what frequency the switcher is running at. I should probably try and analyse it in more detail than I've done so far: that is, poke it briefly with the 'scope and go "ew, that's dirtier than I was expecting". 
Yeah, do that. Then figure how much LC is needed, and if an LDO is still needed to cover the bottom end / save space.
I wouldn't know how to pick an LDO regulator with good PSRR, though. It seems difficult to make comparisons because the manufacturers all give specs according to different conditions. Some give it at 1kHz, some at 100kHz, etc. I'm not even sure what magnitude of PSRR is good. Is 50 dB good or poor? 70 dB? 90 dB?
Oh well they usually give it at DC. So you get numbers in the table like 60, 80, 100dB that are meaningless. But you pop over to the PSRR vs. freq plot (if they give it, and if not, keep shopping!) and see what it's like. Note that it usually falls near zero at high frequencies (control loop GBW, i.e. feedback gain drops below 1 around that point -- it literally can't respond anymore), at which point bypass caps take over, there might be peaks or valleys, it goes back up at HF... this all depends on the fixture used (which they never tell you about), things like stray L in the layout, ESL of the caps, how many caps are used and in what combination (parallel (ceramic) caps have resonance between them), but in any case you can do the same, your LC filter will take over above cutoff (and preferably somewhat below, as needed).
But note that PSRR vs. F is usually done at some large-ish drop, maybe 1V or more. If they give a series of curves with Vdrop as parameter, that's great. Assume it gets worse as Vdrop goes down.
Doing statistics, will be sensitive to AC readings, and you'll want a stable reading to tell more from less. If it's unstable to begin with, at least seeing the variance increase would be a thing, but you can't tell if it's in/out of phase with the system noise floor (which isn't noise at all but coherent: peaks in the frequency spectrum), and any time those correlate, you end up with (constructive/destructive) interference, and you could see the variance decrease instead, which would just be strange, but is perfectly reasonable when things line up just right.
Note that, so far, you've implied these signals are independent, or can be assumed as such. You might know better. If the other supplies are derived from the first, then that ripple carries through according to the transfer functions of everything between, and the statistics will be stable. If they're separate switching circuits, likely not, but there could still be weird ratios and mixing tones that result in fluctuating readings. Gathering statistics over a long enough period, reduces those fluctuations to only when beat frequencies are on the order of that collection period, making such error much rarer, and if the frequencies themselves aren't that stable in the first place (e.g. COT controllers, spread spectrum), the error can also be made small.
Tim
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