An electronic device may have a display such as an organic light-emitting diode (OLED) display. The OLED display may have an array of OLED pixels that each have OLED layers interposed between a cathode and an anode. The pixels may be microcavity OLED pixels having optical cavities. The optical cavities may be defined by a partially transparent cathode layer and a reflective anode structure. The distance between the partially transparent cathode layer and the reflective anode structure for a pixel may be selected such that light at the wavelength emitted by the pixel forms a standing wave between the anode and the cathode. The standing wave may have only one anti-node and the emissive layer for the pixel may be aligned with that one anti-node. To mitigate short circuits, a roughness reduction layer and/or short-circuit-reducing layer having a high sheet resistance may be formed between the anode the OLED layers.