Nanomaterials with quasi-zero, one and two dimensionalities, including silicon nanoparticles, carbon nanotubes, titanium oxide particles and graphene flakes, have been incorporated into the conducting polymer polyaniline to form nanocomposite materials for making flexible supercapacitor sheets. Characterization of the capacitor sheets showed that the inclusion of the nanomaterials in polyaniline has significantly improved the energy and power capabilities of the capacitor. In particular, a specific capacitance of 477.1F/g has been obtained. The important properties of carbon nanotubes of improving the charge storage and reducing the resistance of the nanocomposite material and hence to enhance the power capability of the capacitor sheets have been studied. Stacks of capacitor sheets have been used to power a LED lamp to demonstrate the potential of the nanocomposite-based capacitor sheet in illumination applications.