Transition metal hydroxides have attracted a lot of attention as the electrode materials for supercapacitors owing to their relatively high theoretical capacity, low cost, and facile preparation methods. However, their low intrinsic conductivity deteriorates their high-rate performance and cycling stability. Here, self-supported sheets-on-wire CuO@Ni(OH)2/Zn(OH)2 (CuO@NiZn) composite nanowire arrays were successfully grown on copper foam. The CuO nanowire backbone provided enhanced structural stability and a highly efficient electron-conducting pathway from the active hydroxide nanosheets to the current collector. The resulting CuO@NiZn as the battery-type electrode for supercapacitor application delivered a high capacity of 306.2 mAh g−1 at a current density of 0.8 A g−1 and a very stable capacity of 195.1 mAh g−1 at 4 A g−1 for 10,000 charge–discharge cycles. Furthermore, a quasi-solid-state hybrid supercapacitor (qss HSC) was assembled with active carbon, exhibiting 125.3 mAh g−1 at 0.8 A g−1 and a capacity of 41.6 mAh g−1 at 4 A g−1 for 5000 charge–discharge cycles. Furthermore, the qss HSC was able to deliver a high energy density of about 116.0 Wh kg−1. Even at the highest power density of 7.8 kW kg−1, an energy density of 20.5 Wh kg−1 could still be obtained. Finally, 14 red light-emitting diodes were lit up by a single qss HSC at different bending states, showing good potential for flexible energy storage applications.