Hard wear resistant coatings that are subjected to contact loading sometimes fail because the coating delaminates from the substrate. In this report, systematic finite element computations are used to model coating delamination under contact loading. The coating and substrate are idealized as elastic and elastic–plastic solids, respectively. The interface between coating and substrate is represented using a cohesive zone law, which can be characterized by its strength and fracture toughness. The system is loaded by an axisymmetric, frictionless spherical indenter. We observe two failure modes: shear cracks may nucleate just outside the contact area if the indentation depth or load exceeds a critical value; in addition, tensile cracks may nucleate at the center of the contact when the indenter is subsequently removed from the surface. Delamination mechanism maps are constructed which show the critical indentation depth and force required to initiate both shear and tensile cracks, as functions of relevant material properties. The fictitious viscosity technique for avoiding convergence problems in finite element simulations of crack nucleation and growth on cohesive interfaces allows us to explore a wider parametric space that a conventional cohesive model cannot handle. Numerical results have also been compared to analytical analyses of asymptotic limits using plate bending and membrane stretching theories, thus providing guidelines for interpreting the simulation results.