A new class of non-holonomic constraints is proposed, namely, completely nonholonomic, partially non-holonomic, and truly non-holonomic. The classification is mainly based on the Frobenius condition and the existence of the integral surfaces of the constraints. The intrinsic properties of non-holonomic constraints, such as the reachability property, vary with respect to different classes. In fact, some nonholonomic constraints do not have the reachability property. The path-planning problems for different classes of non-holonomic constraints also become quite delicate. The paths from a point on one side of the critical surface to a point on the other side must pass through the singular points on the critical surface.