A study is made of the competition between failure modes in ceramic-based bilayer structures joined to polymer-based substrates, in simulation of dental crown-like structures with a functional but weak "veneer" layer bonded onto a strong "core" layer. Cyclic contact fatigue tests are conducted in water on model flat systems consisting of glass plates joined to glass, sapphire, alumina or zirconia support layers glued onto polycarbonate bases. Critical numbers of cycles to take each crack mode to failure are plotted as a function of peak contact load on failure maps showing regions in which each fracture mode dominates. In low-cycle conditions, radial and outer cone cracks are competitive in specimens with alumina cores, and outer cone cracks prevail in specimens with zirconia cores; in high-cycle conditions, inner cone cracks prevail in all cases. The roles of other factors, e.g. substrate modulus, layer thickness, indenter radius and residual stresses from specimen preparation, are briefly considered.