We present an extensive study of the quantum dynamics of molecular hydrogen trapped within the nanocavities of two hydrate structures at low temperatures, namely, clathrate structure II and filled ice structure Cl. By inelastic neutron scattering measurements, we investigate a (i) simple H-2 hydrate in clathrate structure II at ambient and high pressure and different temperatures, (ii) binary He-H-2 hydrate in clathrate structure II at high pressure and different temperatures, (iii) simple D-2 hydrate in clathrate structure II at ambient pressure and different temperatures, and (iv) simple H-2 hydrate in structure Cl at high pressure and different temperatures. Molecular quantum rotations and translations, as well as combinations of these are identified in the spectra for hydrogen molecules in the small and large cages of clathrate structure II and in the channels of structure Cl and their energy is provided.