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The efficiencies of dioxygen reduction on common carbonaceous materials were compared using voltammetry and fuel cell measurements. Carbon paper (CP), carbon fibre (CF) or carbon cloth (CC) conducting supports were covered under water pump pressure with multiwalled carbon nanotubes (MWCNTs) to increase the working surface of the electrode, improve connectivity with enzyme molecules and provide direct electron transfer. Laccase was the biocathode catalyst catalyzing 4-electron reduction of oxygen to water on the nanostructured electrode. CP carbon paper was selected as the favourable electrode substrate, since it provided best durability holding firmly the carbon nanotubes together with the enzyme at the electrode surface. Zinc disc or fructose dehydrogenase was used as anode in the hybrid fuel cell and biofuel cell, respectively. The characteristics under externally applied resistances and potential-time dependencies under flowing solution conditions were evaluated. The hybrid fuel cell based on Zn anode and CP supported laccase cathode gave the best results in terms of power and open circuit potential (OCP). The full biofuel cell based on laccase and fructose dehydrogenase shows lower OCP but the power-time dependencies were similar to those of the hybrid biofuel cell. The nanostructured surfaces show good supercapacitor properties due to the presence of carbon nanotubes at the electrode surface. The fuel cells undergo self-charging/discharging and, therefore, can be conveniently employed in pulsed-work regime to power external devices.