Amplitude stabilisation of LC oscillators
Like diode stabilisation and FET stabilisation in RC oscillators, what
methods are used for the stabilisation of amplitude in LC oscillators?

Exactly the same methods. Consider this common-emitter Colpitts Oscillator: -

Transient response showing clear signs of too much loop-gain hence distortion: -

Then consider this modified Colpitts Oscillator with diode clamps: -

Transient response showing very little distortion: -

Images from this website with full explanation of the common-emitter Colpitts Oscillator.