We have carried out inelastic neutron scattering studies of the methane/C-60 system in order to understand the hindered rotational and vibrational dynamics of methane trapped in C-60 interstices and to determine the structure around the interstitial site. At temperatures of 20 K and below, we observe inelastic peaks from rotational transitions of the CH4. These transitions allow unambiguous assignment of the hindered rotational energy levels and a determination of the interaction potential. The appearance of two peaks for one of the J=0 --> 3 transitions implies the existence of two distinct kinds of interstitial sites and the measured transition energies suggest rotational barriers of about 26 and 16 meV for these sites. Time-dependent changes in peak heights indicate slow (t(1/2)approximate to 2.6 h) triplet --> quintet nuclear spin conversion that necessarily accompanies the J=1 --> 0 rotational relaxation. We also have observed an inelastic peak at 10.9 meV in a region where there is a gap in the calculated, and otherwise dense, rotational spectrum. We believe that this must correspond to a local vibrational mode of CH4 rattling in its cage at similar to 2.6 THz. Other peaks involving higher-energy vibrational excitations in CD4/C-60 correspond in energy to assigned peaks in the inelastic neutron scattering spectra of C-60, albeit sometimes with different intensities. Taken together, these findings suggest the rotation of CH4 is hindered by the C-60 cage and that the vibrations of CH4 and C-60 are loosely coupled.