A display may have thin-film transistor (TFT) circuitry on a substrate. An array of organic light-emitting diodes may be formed on the thin-film transistor circuitry. The display may include inorganic brittle layers and organic and metal layers that are ductile and mechanically robust. To help prevent propagation of cracks and other defects along the edge of the display, the display may be provided with crack stop structures and crack detection circuitry. The crack detection circuitry may include one or more loops that are formed along the periphery of the display. The crack stop structures may include TFT/OLED structures formed in a staggered configuration. At least some of the brittle layers can be removed from the panel edge. An additional adhesion layer may also be formed directly on the substrate to help prevent inorganic layers from debonding from the surface of the substrate.