An electronic device such as a voice-controlled speaker device may have a housing characterized by a vertical longitudinal axis. A flexible substrate such as a flexible mesh substrate with component support regions coupled by flexible segments may be wrapped around the housing and the vertical axis. The housing may have surface regions with compound curvature. The flexible substrate may conform to the regions with compound curvature. A fabric spacer layer may be interposed between the flexible substrate and the housing. Electrical components such as input-output devices may be mounted to the component support regions. A display may be formed from an array of light-emitting devices that are mounted on respective component support regions. Light from the light-emitting devices may pass through the fabric spacer layer toward the housing and back out away from the housing. An outer fabric layer may cover the mesh.