Slow magnetosonic shock fronts are unstable to perturbations in their shape or velocity, a phenomenon commonly referred to as a currugation instability. We summarize the results of a linear stability analysis for strong (i.e., high sonic Mach number), adiabatic, slow shocks with arbitrary orientation of the magnetic field with respect to the shock normal. Strong slow shocks in which the field is parallel to the shock normal are unconditionally unstable, with a growth rate which is proportional to the square of the Alfvénic Mach number (to leading order) and is directly proportional to wavenumber. Thus, perturbations grow most rapidly on the smallest scales. For oblique slow shocks, there exists an interval in the inclination angle of the field with respect to the normal in which the shock is stable to perturbations in the plane defined by the upstream and downstream magnetic field. Outside this interval, the front is overstable. However, we show that strong oblique slow shocks are unconditionally unstable to perturbations perpendicular to this plane; thus in three dimensions such shocks are never stable.To follow the evolution of the corrugation instability into the nonlinear regime, we use a time-dependent magnetohydrodynamics code. At early times (i.e., in the linear regime) the numerically measured growth rates are in close agreement with the predictions of linear theory. In the nonlinear regime, we find that parallel slow shocks develop large-amplitude fingers which grow without bound, ultimately destroying the front. Oblique slow shocks develop large-amplitude propagating waves in the plane defined by the magnetic field, but develop fingers in the direction perpendicular to this plane which are similar to the two-dimensional evolution of parallel shocks. Thus, in three dimensions, oblique shocks fragment into sheets, each of which contains overstable oscillations. The corrugation instability studied here is primarily of interest as applied to the accretion shock at the base of magnetically dominated accretion columns in such systems as strongly magnetized white dwarfs and T Tauri stars. We argue that the corrugation instability can result in particle acceleration by the "shock-drift" mechanism. Moreover, we show that the strength of the magnetic field observed in AM Her systems is consistent with the value required for periodic oscillations of the accretion shock driven by the corrugation instability, suggesting that the instability is at least playing a role in producing the variability in these systems.