Stably-stratified layers are present in stellar interiors (radiative zones) as well as planetary interiors (recent observations and theoretical studies of the Earth's magnetic field seem to indicate the presence of a thin, stably-stratified layer at the top of the liquid outer core). We present direct numerical simulations of this region, which is modeled as an ax-isymmetric spherical Couette flow for a stably-stratified fluid embedded in a dipolar magnetic field. For strong magnetic fields, a super-rotating shear layer, rotating nearly 40% faster than the inner region, is generated in the stably stratified region. In the Earth context, and contrary to what was previously believed, we show that this super-rotation may extend toward the Earth magnetostrophic regime if the density stratification is sufficiently large. The corresponding differential rotation triggers mag-netohydrodynamic instabilities and waves in the stratified region, which feature growth rates comparable to the observed timescale for geomag-netic secular variations and jerks. In the stellar context, we perform a linear analysis which shows that similar instabilities are likely to arise, and we argue that it may explain the observed magnetic desert among massive and intermediate mass stars.