Magnetorheological elastomers (MREs) are ferromagnetic particle impregnated rubbers whose mechanical properties are altered by the application of external magnetic fields. In addition, these composite materials can deform at very large strains due to the presence of the soft polymeric matrix without fracturing. From an unconventional point of view, a remarkable property of these materials is that while they can become unstable by combined magneto-mechanical loading, their response is well controlled in the post-instability regime. This, in turn, allows us to try to operate these materials in this critically stable region. These instabilities can lead to extreme responses such as wrinkles (for haptic applications), actively controlled stiffness (for cell-growth) and acoustic properties with only marginal changes in the externally applied magnetic fields. Unlike the current modeling of hierarchical composites, MREs require the development of finite-strain coupled nonlinear magneto-mechanical models in order to tailor the desired macroscopic instability response at finite strains. As a proof of concept, we study experimentally and theoretically the stability and post-bifurcation of a non-linear magnetoelastic film/substrate block in order to obtain active control of surface roughness. The non-intuitive interplay between magnetic field and elastic deformation owes to material and geometry selection, namely a ferromagnetic particle composite film bonded on a compliant passive foundation. Cooperation of two otherwise independent loading mechanisms–mechanical pre-compression and magnetic field–allows to bring the structure near a marginally stable state and then destabilize it with either magnetic or mechanical fields. We demonstrate for the first time that the critical magnetic field is a decreasing function of pre-compression and vice versa. The experimental results are then probed successfully with full-field finite element simulations at large strains and magnetic fields. The magnetoelastic coupling allows for the reversible on/off control of surface wrinkling under adjustable critical magnetic and mechanical fields. In this view, this study constitutes a first step towards realistic active haptic and morphing devices. Novel auxetic and chiral architected MREs are also proposed as potential candidates for future work.