What type of oscillator circuits are used in Capactive proximity sensors?
There are plenty of capacitance probes but I feel compelled to explain the type that I have been involved with.

The type of oscillator was based around a common-collector colpitts circuit and used an operating frequency of around 10 MHz. Using this frequency you get a decent signal when there is a high speed situation to be detected such as on the blades of a turbine passing at full speed. The 3 dB bandwidth was up to 70 kHz from memory.

The down side is that at 10 MHz, transmission line effects came into play so this was utilized to good effect; a changing capacitance at the end of the probe became a changing inductance at the oscillator terminals and frequency was modulated as metal passed-by the probe end.

More often than not an LC oscillator is used (in my types of application) but I can't rule out that an RC oscillator design is used in some other applications.