Positron annihilation lifetime and Doppler-broadening energy spectra have been measured between 13 and 95 K for 0, 0.50, and 0.81 submonolayers of methane adsorbed on graphite. Both the surface positron lifetimes (\ensuremath{\sim}0.31 ns) and S parameter (central) of the Doppler-broadening energy spectra are sensitive to the solid-solid, solid-liquid, and liquid--hypercritical-fluid phase transitions. Activation energies of 0.017\ifmmode\pm\else\textpm\fi{}0.002 eV and 0.007\ifmmode\pm\else\textpm\fi{}0.002 eV were obtained for surface-defect formation for n=0.5 and 0.81, respectively, in two-dimensional methane on graphite surfaces. Applications of positron annihilation spectroscopy to study physisorption phenomena are discussed.