We report high-resolution scanning tunneling microscopy and spectroscopy of hydrogenated, quasi-free-standing graphene. For this material, theory has predicted the appearance of a midgap state at the Fermi level, and first angle-resolved photoemission spectroscopy (ARPES) studies have provided evidence for the existence of this state in the long-range electronic structure. However, the spatial extension of H defects, their preferential adsorption patterns on graphene, or local electronic structure are experimentally still largely unexplored. Here, we investigate the shapes and local electronic structure of H impurities that go with the aforementioned midgap state observed in ARPES. Our measurements of the local density of states at hydrogenated patches of graphene reveal a hydrogen impurity state near the Fermi level whose shape depends on the tip position with respect to the center of a patch. In the low H concentration regime, we further observe predominantly single hydrogenation sites as well as extended multiple C-H sites in parallel orientation to the lattice vectors, indicating an adsorption at the same graphene sublattice. This is corroborated by ARPES measurements showing the formation of a dispersionless hydrogen impurity state which is extended over the whole Brillouin zone.