The aim of this study is to highlight nonlinear behaviors and periodic orbits of the single-track vehicle model with a delayed feedback controller. Two widely used tire models, namely a linear tire characteristic and Pacejka’s Magic Formula are considered. Linearly stable domains of parameters such as the vehicle speed and the control gains are determined. Periodic solutions originating from Hopf bifurcation points are followed using numerical continuation and the results obtained with the two different tire models are compared. It is shown that neglecting the saturation of the tire lateral forces at total sliding might change the sense of certain Hopf bifurcations from subcritical to supercritical. The results are verified by numerical simulations. The resulting bifurcation diagrams aim to quantify the degree of robustness of these controllers with regards to the initial conditions at various parameter ranges in order to assure stable and safe operation.