In order to study the influence of different operating modes on the dynamic response and stability of offshore wind turbines under turbulent wind and earthquake, DTU 10 MW offshore single-pile wind turbine was used as the research object. The p-y curve was used to describe the interaction between the pile foundation and the subsea soil. Based on the finite element software ANSYS, the finite element model of the wind turbine was established and the structure modal, dynamics and buckling calculation of the wind turbine were carried out. The results show that the vibration mode of the tower is dominated by torsional and bending vibration. Its first natural frequency is within the ideal design range and the rotation of the wind wheel will not cause its resonance. Turbulent wind has the most significant influence on the displacement of tower top and earthquake has much greater influence on the lateral displacement of tower top than the turbulent wind. After the earthquake, the equivalent stress and strain energy response of the tower have surged. The emergency shut-down can effectively reduce the peak of the tower top displacement, the tower equivalent stress and the tower strain energy response and reduce the risk of structural damage and plastic deformation. Under the action of turbulent wind and earthquake, lateral buckling failure will occur in the middle and upper part of the leeward surface of the tower. The emergency shut-down can effectively improve its buckling resistance and the maximum increase of critical buckling load is nearly 27.11%.