Using quantum Monte Carlo we have studied the superfluid density of the first layer of 4He and H2 adsorbed on graphene and graphite. Our main focus has been on the equilibrium ground state of the system, which corresponds to a registered √3 × √3 phase. The perfect solid phase of H2 shows no superfluid signal, whereas 4He has a finite but small superfluid fraction (0.67%). The introduction of vacancies in the crystal makes the superfluidity increase, showing values as large as 14% in 4He without destroying the spatial solid order.