The proton spin-lattice relaxation times and proton chemical shifts for the hydroxyl and methyl protons in methanol were measured at liquid and supercritical densities using capillary high-pressure NMR spectroscopy. The pressure range for the proton nuclear relaxation measurements was between 50 and 3500 bar over a temperature range of 298-573 K. The proton chemical shifts of methanol were investigated for a pressure range of 50-3500 bar and a temperature range of 298-773 K. Attempts were made to separate the contributions of the dipolar and spin-rotation interactions to the spin-relaxation processes at each thermodynamic condition over methanol densities ranging from liquid to supercritical fluid. An average number of hydrogen bonds per molecule in methanol and the apparent activation energy of the methyl group internal rotation have been extracted from the experimental relaxation data. The extracted quantities show a moderate pressure dependence in addition to temperature effects, which suggest that molecular packing effects on hydrogen-bonded methanol are important at higher pressures. A comparison between methanol and water at similar thermodynamic conditions was also made to obtain new insight into these two important supercritical solvents.