The paper deals with the bifurcation analysis of a rather simple mathematical model describing an automobile running on an even surface. The mechanical model has two degrees of freedom and the related equations of motion contain the nonlinear tyre characteristics. An experimental validation of the model is produced. Additionally the presented results are cross-validated by exploiting handling diagram theory. Bifurcation analysis is adopted as the proper procedure for analysing steady-state motion (either cornering or running straigh ahead). Two independent parameters referring to running conditions, namely steering angle and speed, are varied. Ten different combinations of front and rear tyre char-acteristics (featuring understeer or oversteer automobiles) are considered for the bifurcation analysis. Many different dynamical behaviours of the model are obtained by slightly varying the parameters describing the tyre characteristics. Both simple and extremely complex bifurcations may occur. Homoclinic bifurcations, stable and unstable limit cycles (of considerable amplitude) are found, giving a sound base of knowledge to control engineers who are asked to implement robust algorithms to reach stability. A bifurcation analysis is also performed referring to straight ahead motion. In this case the driver (either human or not) is included in the loop. Also in this case limit cycles exist that may suggest how complicated is controlling the stability of such relatively simple running condition. The knowledge of the derived set of bifurcations is dramatically important to fully understand the actual vehicle yaw motions occurring while running on an even surface.