Just powered the board using my regulated bench top supply and it has the exact same oscillations. So it must be created on the board. What I find strange is that the 3.3V is clean as a whistle but the 24V and 12V are not.
All linear regulators are not created equal. A lot of the older ones, often named 78*XX, usually attenuates 50/60/100/120 Hz ripple but higher frequencies will pass through. LC-filters before and in between the regulators might help. Another thing you can try is to swap the ceramic capacitors on the outputs of the 78** to something with a little bit more ESR, for instance tantalum or ceramic cap with a 1ohm series resistor. I found no info the datasheet for the 78** from OnSemi, but it might be sensitive to too low ESR on the output.
You also asked about increasing the sampling time in a previous post. Give it a try and see if it gives any improvement. If you have an oscilloscope, try to set the sampling time longer than the dip on the ADC input.