When wheels are rigid relative to other parts of the supporting structure, complex shimmy phenomena can be described by single-contact-point models and the equations of motion of the corresponding nonholomic systems can be derived by means of Appell-Gibbs equations. Due to the simple geometric nonlinearities of these systems, the arising Hopf bifurcations are typically subcritical, and even isolated unstable periodic motions may exist in the parameter space. The prediction of the isolated unstable self-excited vibrations is difficult at the design stage, and even extensive experiments may fail to identify them. This becomes even worse when Coulomb friction exists at the king pin of these structures, which is modeled as another strong nonlinearity in the system. Analytical studies extended with AUTO-07p numerical bifurcation analysis of such systems are carried out to construct bifurcation charts while the rolling condition is also checked for the contact force between the wheel and ground. Parameter domains are identified, where bistability occurs, that is, where stable stationary rolling and violent shimmy may coexist. The results also support engineers to select the most relevant parameters, the variations of which can effectively help to reduce the vibration amplitude of shimmy or to increase the critical towing speeds where shimmy may appear.