Issues of stability and bifurcation phenomena in vehicle's steering dynamics are presented. Based on the assumption of constant driving speed, a second order nonlinear lateral dynamical model is obtained. Local stability and existence conditions for saddle node bifurcation appearing in vehicle dynamics with respect to the variation of the front wheel steering angle are then derived via system linearization and local bifurcation analysis. Bifurcation phenomena occurring in nonlinear vehicle lateral dynamics might result in spin and/or system instability. Perturbation method is employed to solve for an approximation of system equilibrium near the zero value of the front wheel steering angle, which reveals the relationship between sideslip angle and the applied front wheel angle. Numerical study of a simple example verifies the theoretical analysis.