Adhesive contact of a rigid sphere with a layered medium consisting of a stiff elastic layer perfectly bonded to an elastic-plastic substrate is examined in the context of finite element simulations. Surface adhesion is modeled by nonlinear spring elements obeying a force-displacement relation governed by the Lennard–Jones potential. Adhesive contact is interpreted in terms of the layer thickness, effective Tabor parameter (a function of the layer thickness and Tabor parameters corresponding to layer and substrate material properties), maximum surface separation, layer-to-substrate elastic modulus ratio, and plasticity parameter (a characteristic adhesive stress expressed as the ratio of the work of adhesion to the surface equilibrium distance, divided by the yield strength of the substrate). It is shown that surface separation (detachment) during unloading is not encountered at the instant of maximum adhesion (pull-off) force, but as the layered medium is stretched by the rigid sphere, when abrupt surface separation (jump-out) occurs under a smaller force (surface separation force). Ductile- and brittle-like modes of surface detachment, characterized by the formation of a neck between the rigid sphere and the layered medium and a residual impression on the unloaded layered medium, respectively, are interpreted for a wide range of plasticity parameter and maximum surface separation. Numerical results illustrate the effects of layer thickness, bulk and surface material properties, and maximum surface separation (interaction distance) on the pull-off and surface separation forces, jump-in and jump-out contact instabilities, and evolution of substrate plasticity during loading and unloading. Simulations of cyclic adhesive contact demonstrate that incremental plasticity (ratcheting) in the substrate is the most likely steady-state deformation mechanism under repetitive adhesive contact conditions.