Struggling - Schmitt Trigger Oscillator

In the morning I commenced the calibration of the frequency counter. It works now so it time to make it work. I picked a breadboard an put together a 120kHz RC oscillator based on a HCMOS Smitt Trigger:

It started, I measured it, the frequency was fluctuating in both directions with 20%. I know that a base RC oscillator is not the champion of the precision and stability, but I can't believe that the result can be such bad.
I was changing the ratio of the R and C values, even lowered the frequency to around 30kHz, it not stabilized.
Reorganized the power supply by putting the oscillator and the counter into a common USB rail. The result is a fluctuating 600kHz. This really stinks - Oscilloscope:

This is the rising edge of the signal.
With 50ns timebase.
It was known to me that the high frequency behavior of the breadboard is a catastrophic, but it was a surprise to me, that I can't put together a 120kHz oscillator on it.
It was in my head that a parallel ceramic capacitor in the feedback loop may slow down the circuit a little bit. My colleague who is a radio frequency expert confirmed it. But this comes tomorrow...