How stable are PID loops really?
There's no inherent oscillation in the output. A perfect PID controller controlling an ideal 2nd order system has a closed-loop transfer function that either moves asymptotically to the setpoint (damping factor greater than 1) or oscillates about the setpoint with the oscillations decreasing to an arbitrarily low value over time (damping factor less than one). Typical tuning is such that the oscillations decrease by a factor of 4 or more every cycle so they would quickly fall below any realistic noise level (\$0.25^{20} \approx 10^{-12}\$). 

Real PID controllers can have limit cycles (for example if there is some stiction in a valve actuator) or they can be time-proportioning types that have an output PWM that shows up in the output. Probably other stuff too. Anything you look at with a 24-bit or better ADC probably has some pattern you can find, even if it originates in the ADC itself (eg. idle tones).