This paper concerns the small-time stabilization of some classes of mechanical systems which are not stabilizable by means of time-invariant continuous state feedback laws. This is the case of nonholonomic systems, an example being the "unicycle-like" mobile robot, or for underactuated mechanical systems, an example being the slider. Explicit time-varying feedback laws leading to small-time stabilization are constructed for these two control systems. The main tools are homogeneity, backstepping, and desingularization techniques.