A geometrical theory of general nonholonomic mechanical systems on fibred manifolds and their jet prolongations, based on so-called Chetaev-type constraint forces, was developed in 1990s by Krupková. The relevance of this theory for general types of nonholonomic constraints, not only linear or affine ones, was then verified on appropriate models. Frequently considered constraints on real physical systems are based on rolling without sliding, i.e. they are holonomic, or semi-holonomic, i.e. integrable. Moreover, there exist some practical examples of systems subjected to true (non-integrable) nonholonomic constraint conditions. On the other hand, the equations of motion of a bicycle are highly nonlinear and rolling of wheels without slipping can only be expressed by nonholonomic constraint equations. In this paper, the geometrical theory is applied to the abovementioned mechanical problem using the above mentioned Krupková approach. Both types of equations of motion resulting from the theory-deformed equations with the so-called Chetaev-type constraint forces containing Lagrange multipliers, and reduced equations free from multipliers are found and discussed.