Reducing noise in microphone amplifier circuit at higher gains
OK, from comments:
That's not noise, that's either instability or interference, perhaps on the PSU. (The regularity of the waveform shows this. Start by arranging a perfectly clean 4.5V supply for the mic amp, and see how it goes.
Having seen that a separate battery supply improves performance, I'd suggest improving the quality of power fed to the mic amp. Simplest approach would be an R-C low pass filter : 100 ohms would drop a fraction of a volt at the current required, and 1000uF (giving an 0.1 second time constant) should attenuate the power supply noise adequately. These values can then be tuned for size, cost, headroom, attenuation to suit your goals.
Andy also deserves credit for suggesting an improvement in opamp selection.