Our contribution in this article is twofold. First, we identify the regularity properties of the trajectories of planar wheeled mobile robots. By regularity properties of a trajectory we mean whether this trajectory, or a function computed from it, belongs to a certain class Cⁿ (the class of functions that are differentiable n times with a continuous n^th derivative). We show that, under some generic assumptions about the rotation and steering velocities of the wheels, any non-degenerate wheeled robot belongs to one of the two following classes. Class I comprises those robots whose admissible trajectories in the plane are C¹ and piecewise C²; and class II comprises those robots whose admissible trajectories are C¹, piecewise C² and, in addition, curvature-continuous. Second, based on this characterization, we derive new feedback control and gap-filling algorithms for wheeled mobile robots using the recently-developed affine trajectory deformation framework.