Methane hydrate (MH) and argon hydrate (AH) single crystals were synthesized in a diamond anvil cell to investigate their intrinsic high-pressure properties by visual observation and in situ Raman and Brillouin-scattering measurements. Single crystalline MH shows clearly two phase transitions at P =0.9 and 1.9 GPa in its crystalline shape and state under a microscope, and in the change in its Raman spectra of CH stretching vibrations of guest CH4 molecules. The first determination of the elastic properties in MH sI phase has revealed that MH is about 10% softer than the most common ice Ih, which is accounted for by MH's void-rich and open structures. For a single-crystal AH, we have observed two phase transitions from AH-I (sII) to AH-II at about P =0.65 GPa and from AH-II to AH-III at P =1.02 GPa. The breakdown of the cage structure in AH-III was confirmed by the disappearance of the OH stretching mode of host H2O lattices. The possibility of Ar double occupancy in the large cages of sII and higher pressure phases is investigated by the low-frequency Raman peak observed around 120 ~ 140 cm1 in view of recent MD calculations. PACS Nos.: 62.30+d, 62.50+p, 62.65+k, 64.70-p, 78.30-j, 78.35+c