Thanks for the reply. I've tried it with several different power supplies, ranging from 9V-12V and up to 2.5A. There is a voltage regulator on the board which outputs 5V and is rated at 1.5A; the MPN is OKI-78SR-5/1.5-W36H-C.
I initially wondered if the noise could be caused by this voltage regulator (it is a switching regulator which I know can be noisy), but I tested it with a linear regulator (MC78M05CDTRKG) and the noise persisted.
I then tried removing the regulator entirely, hacking a USB cable and soldering the power line directly to VIN_USB (P1.07) on the PocketBeagle, both with and without a ferrite bead - the latter of which I assumed should be exactly the same as plugging in a USB cable to the PocketBeagle, but again the noise persists.
I have also tried powering the Bela from both the regular VIN (P1.01) and the VIN_USB (P1.07) pins, which doesn't change anything.
In answer to your other question, yes I am using all 8 of the analogue inputs to read sensors, although when I comment this out the noise is still present.