Wow! Are you are going MIL spec then? I’m just guessing by the environmental tests. High temp, vibration, etc.
On a serious note, it might indicate a power supply problem. My first ever project from ‘Practical Electronics’ was a 3 transistor (germanium!) amplifier. It made all kinds of weird ‘motor boat’ noises until I shunted the dry battery I was using with an electrolytic. My first lesson in power supply stability.
Looking at the noise in ‘Audacity’, and ‘blowing it up’, I notice a background 10mS rough half sine wave, plus the HF noise. You’re coming out of regulation, unless you’re pulling more than (say) 150mA, so I’d try a series inductor in the 12V make sure you have local decoupling capacitors on all your boards. You should have both a small electrolytic with a ceramic in parallel, (Electrolytic don’t work too well at high - like PIC clock - frequencies). All in all my vote is for the PSU as a source of the noise. From the description, I’d say your supply was a switch mode. I prefer a linear for most audio kit. There are usable switch modes, (I’ve just bought an old Akai 3000 sampler that uses one), but they are ‘instrumentation grade’ and expensive.
Try loading the supply with resistors - suggest 4.7Ohm on the 5V and 150 Ohm on the 12 V rails, (get big ones 10 watt for the 4.7 and 2 watt for the 150’s), and then put a scope across the power rails, on AC coupling, an see what you get. I use a test like this often, before trusting the supply with the expensive stuff. I’ve just done something similar with both the Oakley 3030 I’ve just completed, (welll all bar 1 LED), I put the regulator chip in before any others and set and tested the power supplies before inserting the expensive stuff. With the AVRx board I’ve just started, I built the PSU section first, out of the normal order, both to make sure it worked, and as a first test of my surface mount skills. With surface mount a failure is usually a lot more expensive…
Perhaps as an overall hint to MIDIboxers in general. A couple of bench power supplies, either second hand, or kits to save money, are a really worthwhile investment, second only to a test meter. I say a couple because you are going to need at least a 5V and the analogue power.