In this paper, we study the phase and amplitude instability mechanisms of mid-infrared quantum-cascade lasers with Fabry-Perot cavity. The phase instability exhibits the single-mode nature and the amplitude instability is the multi-mode Risken-Nummedal-Graham-Haken kind of instability. The stability analysis is performed based on the nonlinear Maxwell-Bloch formulism. The phase instability and amplitude instability are decoupled based on the symmetry and anti-symmetry of propagating modes in the cavity. We also discuss the influences of Kerr nonlinearity from optical transitions on these instability mechanisms. The Kerr nonlinearity exhibits as the combination of saturable absorber effect in the transverse direction and the self-phase modulation in the longitudinal direction. The saturable absorber could lower the threshold of both instability mechanisms. The self-phase modulation has impact on the instability spectra of quantum-cascade lasers. It could not only broaden the unstable frequency domain, but also transform it from the single-mode to the multi-mode domain.