An electronic device such as a speaker device may have a curved housing characterized by a vertical longitudinal axis. A layer of fabric may cover the curved housing. A touch sensor may be used to detect touch input on the layer of fabric. The touch sensor may include capacitive touch sensor electrodes including drive lines and sense lines. In some arrangements, the touch sensor is formed from conductive strands in the layer of fabric. In other arrangements, the touch sensor is formed from conductive traces on a substrate. The substrate may be formed from portions of the curved housing or may be formed from a layer that is separate from the housing. Light-emitting components and/or fabric with different visual characteristics may be used to mark where the touch-sensitive regions of the fabric are located. The touch-sensitive regions may be shaped as media control symbols.