This contribution is concerned with theoretical and numerical aspects of instability phenomena in multisurface elasto-plasticity which is the adequate constitutive framework for e.g. single crystal plasticity. To this end, the localization analysis for multisurface elasto-plasticity based on the additive decomposition of the geometrically linearized strain tensor is considered and the characteristic differences and similarities to the wellknown single surface case are highlighted. In the numerical part we subsequently discuss the computational setting of single crystal plasticity as the paradigm for multisurface elasto-plasticity. Finally, the numerical examples demonstrate the dramatic influence of the slip system orientation on the resulting localized failure mode of a single crystal compression panel.