The free energy contained in electron drift, electron collision, and gradient of plasma density, temperature, magnetic can trigger the different frequency and wavelength instabilities in hall thrusters. The instabilities will destroy the stable discharge of plasma, affect the matching degree between the thruster and the power processing unit, and down the performance of the thruster. Based on this, the instabilities triggered by electron collision and gradient of plasma density and magnetic field in the hall thruster is studied by using dispersion relation derived from the fluid model. The results show that: 1) The instabilities can be excited at any axial position from the near anode region of the thruster to the plume region when the effect of electron inertia、electron collision with neutral atoms and electron drift are included in the model. The transition of the lower-hybrid mode excited by electron collision into the ion sound mode take place with the azimuthal wavenumber k_yis increasing. Where k=2 π/ λ， λis the wave length. The real frequency ω_rcorresponding to the maximum growth rate γ _maxslightly decreases with collision frequency increasing for k_y=10m ^-1. However, the maximum real frequency and real frequency ω_rcorresponding to the maximum growth rate γ _maxwill not change with collision frequency varying for k_y=300m ^-1. Independent on the size of k_y, the growth rate of mode triggered by electron collision increases with collision frequency increasing. 2) The plasma density gradient effect plays the dominant role in driving instabilities when the electron inertia, electron-neutral collisions and plasma density gradient are simultaneously included in the model. The dynamic behavior of the model does not change with the increasing of k_y, but the eigenvalue of the model increases with the k_yincreasing. Since the sign of anti-drift frequency induced by the plasma density gradient is changed, and the mode eigenvalue have the opposite change trend on both sides of point κ_N=0. When the sign of ω_sand ω_rare opposite, the density gradient effect has a stabilization effect on instability excitation ( κ_N＞0). When the sign of ω_sand ω_rare same, the density gradient effect enhances the excitation of instability ( κ_N＜0) .3) If the gradient of the plasma density and magnetic field, electron inertia and electron-neutral collisions are included in the dispersion, the mode eigenvalue relies on the size of electron drift frequency, and the diamagnetic drift frequency induced by the gradient of density and magnetic field. When the density gradient and magnetic gradient effect are considered, there is a stable window in the discharge channel. However，if the electron inertia and electron-neutral collisions are also included, the stable window will disappear.