When two bodies interact in contact, self-oscillations or relaxation vibrations can occur, which are vibrations of parts of bodies relative to each other. The study of the processes occurring in the zone of an elastic tire interaction with a solid support base is of particular interest, since they directly affect the safety of movement of wheeled vehicles. The aim of the work is to study the conditions for the occurrence of self-oscillating processes in the zone of interaction of an elastic tire with a solid support base in three rolling modes: traction, driven, and braking. It has been established that the appearance of a self-oscillating mode in the zone of interaction of an elastic tire with a solid support base is a useful diagnostic feature providing recognition of the development of the process of losing adhesion of a wheel with a support base when the rear axle skids at early stages.