It looks like troubles with the PSU. Just to confirm, why don't you make one of the analog outputs output a constant voltage and connect it to your input, just to make sure you do not see any of those spikes?
I also like to "listen to the PSU": connect the PSU to an AC-coupled audio input (maybe put a few hundred ohms in series, just in case) and listen to it. The noise you hear there will leak into your signal.
You do not really need a PSU for this application, a voltage reference is all it takes. A few silicon diodes in series or a 4V zener would probably give you a stable enough reference (remember the analog input range is 0-4.096V), alternatively, look for a proper voltage reference IC. Also an LM7805 would probably be good enough for this application, but voltage references are less noisy and you do not need much current, so they are better suited. Note that stabilizers and references have up to 2V of dropout, so you need either a PSU with higher voltage or pick a low-dropout device. You can also try if you are lucky using one of the voltages available on the BBB, like the 3.3V on P9-3 and P9-4 (you will have to change your upper resistor in the divider to match the new voltage).
I have no recommendation about commercially available power supplies... but prices can be quickly become esoteric if you google for "audio power supply", at which point you are probably better off with a transformer, bridge rectifier, cap and LM78xx ...