Simulations of the magnetohydrodynamic (MHD) dynamics of a current-vortex sheet for low values of the Alfvén number (A = VA/V0) in both the subsonic and transonic regimes are described. It is shown that, in agreement with previous linear results, the shear layer is unstable for all values of the sonic Mach number (M). However, the subsonic and transonic disturbances which develop differ significantly in many of their properties. In the subsonic case the evolution of the system is similar to the incompressible one since the growing perturbation is symmetric and evanescent in the cross stream direction. The inclusion of compressible effects mainly permits the study of its thermodynamic behavior. In the transonic case the growing perturbation appears to be overstable, asymmetric, oscillatory, and weakly evanescent in the cross stream direction. In their nonlinear evolution such modes, which we identify as fast MHD waves, lead to the formation of shocks and to a very different dynamics of the current-vortex system.