For analog, electret microphones a mute switch is very simple. Simply put an R-C pair (R a few MΩ, C 10-20 µF) in parallel to the microphone capsule. To mute it, short the R out. This is pop-free because the R charges the capacitor to the DC operating point of the capsule. When the R is shorted out, the microphone's output is shunted by the capacitor, which attenuates the signal level by >40-60 dB.

For digital microphones it can also be very simple. Consider a PDM microphone, if you disconnect the data line and have it stick to either high or low through a pull-up, the signal becomes DC. This wouldn't be pop-free by itself, but it should be rather easy to make it so through DSP.

(FWIW, you can also easily mute pro-audio microphones by shorting (+) and (-) together. The attenuation is determined by the output impedance of the microphone, usually a few hundred Ohms, divided by the resistance of your switch. Typically you'll get like 60 dB from this, which is perfectly fine for muting in a normal setup, but if you crank the preamp all the way up you can still get a usable, albeit very noisy, signal out of it.)