When LNA’s S11 is positive in dB at some frequencies is not necessarily to oscillates, but at that frequencies is potential unstable. The reason could be the layout design (external feedback), internal feedback inside the active device, the emitter (source) degeneration, the load presented at the output, excess gain at that frequencies, etc.
A better characterization of instability is K-factor (Rollet Stability Factor). When K-factor is greater than unity, the circuit will be unconditionally stable for any combinations of source and load impedance.
Another stabilization method can be realized with a simple emitter feedback inductor. A small inductor can make the circuit more stable at higher frequencies. But if the source inductance is increased, the K-factor at higher frequencies eventually falls bellow 1. This effect limits the amount of source inductance that can safely be used.