Somehow, I thought you'd already corrected that. It's a bad idea to overload the USB power supply, especially if it's a USB port on your computer! You wouldn't want to fry that!
That's not going to be practical. You'd have to wait for the voltage to stabilize every time the LEDs change and you'd have to have an adjustment factor for all of the LED variations. Just get a better (higher current) power supply. You may still have some noise issues but you should start with a good power supply.
What was happening was the LED load was putting a square wave component onto the power rail due to the large currrents and resistance of the wires and breadboard contacts (these can be quite large BTW).
That microphone module clearly had inadequate power-supply-rejection (aka PSRR), and was amplifying those steps in the power rail by a large factor, but only at AC (hence the recovery after a 100ms as DC conditions re-establish themselves).
Keeping the LED power entirely separate except for one ground-common point is the main fix (I think you have that now), but also realize that the mic module is a poor performer for PSRR and needs strong decoupling to work well.
[ From what I can see of the markings that mic module is a knock-off, not a genuine Adafruit module, and probably has counterfeit parts on it, or may even simply lack decoupling - knock-off modules are often low quality like this ]
The part is an original. I need to run of a single USB power source, not a computer port but a standard 2.1A rated charger or power-bank.
So it is! 
Didn't notice it as it showed up so small. Anyway, point made. 
Well in that case its a different silk-screen from the pictures at Adafruit - they moved some of the front-side printing.
That MAX4466 chip isn't all its touted to be in the datasheet(*), its extremely noisy for instance quoting 80nV/√Hz rather than a standard audio opamp which mostly have 3--15nV/√Hz range (~15 to 30dB better!!!!!) It claims 80dB PSRR which is probably just enough.
I suspect a standard rail-to-rail opamp circuit will be much better (much less noise) anyway and have decent PSRR. Sparkfun do an OPA344 breakout (still poor at 20nV/√Hz, but 8.5dB better than the MAX4466 https://www.sparkfun.com/products/12758
(*) Its specifically a high noise preamplifier for microphones, but datasheets can use "low" whenever they like as an adjective has no meaning in this context in law.
To finish this topic up:
After some back and forth I am using the attached circuit now. It's an electret mic with no amp - for the purpose of picking up sound it works well enough.
Interestingly the amps drawn by the LEDs do not play a noticeable roll.
USB power is notoriously noisy, clear up into the radio spectrum. You need a decent power supply filter, and shielded cable for sensitive electronics.
this picture looks like left half of Adafruit Mic Module schematic
pretty much, slightly different R and C
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