Mass production of high-quality graphene sheets is essential for their practical application in electronics, optoelectronics, composite materials and energy-storage devices. Here we report a prompt electrochemical exfoliation of graphene sheets into aqueous solutions of different inorganic salts ((NH4)¬2SO4, Na2SO4, K2SO4, etc). Exfoliation in these electrolytes leads to graphene with a high yield (>85%, ≤3 layers), large lateral size (up to 44 μm), low oxidation degree (a C/O ratio of 17.2), and a remarkable hole mobility of 310 cm2 V-1 s-1. Further, highly conductive graphene films (11  sq.-1) are readily fabricated on an A4-size paper by applying brush painting of a concentrated graphene ink (10 mg/mL, in N,N'-dimethylformamide (DMF)). All-solid-state flexible supercapacitors manufactured based on such graphene films deliver a high area capacitance of 11.3 mF cm-2 and an excellent rate capability of 5000 mV s-1. The described electrochemical exfoliation shows great promise for the industrial-scale synthesis of high-quality graphene for numerous advanced applications.