Simple explicit relations are presented for the onset of competing fracture modes in ceramic coatings on compliant substrates from Hertzian-like contacts. Special attention is given to a deleterious mode of radial cracking that initiates at the lower coating surface beneath the contact, in addition to traditional cone cracking and quasiplasticity in the near-contact area. The critical load relations are expressed in terms of well-documented material parameters (elastic modulus, toughness, hardness, and strength) and geometrical parameters (coating thickness and sphere radius). Data from selected glass, Al2O3 and ZrO2 coating materials on polycarbonate substrates are used to demonstrate the validity of the relations. The formulation provides a basis for designing ceramic coatings with optimum damage resistance.