An electronic device may be provided with a display. The display may be mounted in a housing. Electronic components may be mounted in an interior region of the device. The display may have an array of pixels configured to display an image. A border region such as a ring-shaped border may run along the outermost peripheral edge of the array of pixels. The display may be covered by a transparent display cover layer. The transparent display cover layer may have opposing inner and outer surfaces. The inner surface may face the interior region of the device. A laser-marked light-scattering structure may be embedded within an interior portion of the display cover layer between the inner and outer surfaces. The light-scattering structure may be located in the border region and may be illuminated by a light-emitting device.