An LC-unstoppable noise in a USB-powered device
The ADC input range is about 2 V, I wanted 10 bits. Yet the analog signal turned out to be so weak that, as seen in the schematic, I needed to set the opamp gain to 40x.

It translates to an expected differentation of input levels of about 2000 mV / 40 / 1024 = 0.05 mV. Not realising at first, that it is such a small value, I did not shield the analog source.

The digital signal is quite clean when the analog source is cut off near the opamp, so the 1 MHz power line noise probably does not affect a lot the sampling at a much lower frequency of 8 kHz. I removed the voltage stabiliser, as it made the analog part work a bit worse and otherwise had no function. I kept NCP380LSNAJAAT1G + the electrolytic condenser, though, as they do remove some low frequency noise, as described in the question.