On a long path of finding appropriate materials to store hydrogen, graphene and carbon nanotubes have drawn a lot of attention as potential storage materials. Their advantages lie at hand since those materials provide a large surface area (which can be used for physisorption), are cheap compared to metal hydrides, are abundant nearly everywhere, and most importantly, can increase safety to existing storage solutions. Therefore, a great variety of theoretical studies were employed to study those materials. After a benchmark study of different van-der-Waals corrections to Generalized Gradient Approximation (GGA), the present Density Functional Theory (DFT) study employs Tkatchenko–Scheffler (TS) correction to study the influence of vacancy and Stone–Wales defects in graphene on the physisorption of the hydrogen molecule. Furthermore, we investigate a large-angle (1,0) grain boundary as well as the adsorption behaviour of Penta-Octa-Penta (POP)-graphene.