An organic light-emitting diode display may have thin-film transistor circuitry formed on a substrate. The display and substrate may have rounded corners. A pixel definition layer may be formed on the thin-film transistor circuitry. Openings in the pixel definition layer may be provided with emissive material overlapping respective anodes for organic light-emitting diodes. A cathode layer may cover the array of pixels. A ground power supply path may be used to distribute a ground voltage to the cathode layer. The ground power supply path may be formed from a metal layer that is shorted to the cathode layer using portions of a metal layer that forms anodes, may be formed from a mesh shaped metal pattern, may have L-shaped path segments, and may include laser-deposited metal on the cathode layer. Data lines may be formed from metal layers in the active area to accommodate the rounded corners of the display.