We report on laboratory measurements of compressional and shear wave speeds in compacted, polycrystalline sI methane and sII methane-ethane hydrates and ice Ih. The hydrate samples were made from granulated ice warmed to 290 K in the presence of methane or methane-ethane gas at high pressure. The resulting porous gas hydrate samples were uniaxially compacted within the synthesis pressure vessel using a hydraulic ram with a moving piston and fixed end plug fitted with shear transducers. Once the samples were fully compacted, the temperature was cycled in steps from 258 to 288 K while the uniaxial pressure was held constant at 60 MPa. After temperature cycling was completed, the uniaxial pressure was varied between 30 and 90 MPa at 283, 273, 263 and 253 K. At the end of each experiment, the uniaxial pressure was slowly decreased to 1 atm at 253 K. Shear and compressional wave speed measurements were made throughout each experiment. For ice Ih, the sample was evacuated before compaction, the measurement temperature range was 253 to 268 K and the applied uniaxial pressure did not exceed 42 MPa. Analysis of the data produces several interesting observations. Among them are: 1) sI and sII gas hydrate resist compaction much more than ice. A pressure of 42 MPa fully compacted the ice sample at 268 K, but a pressure of 105 MPa had to be applied for several days (at temperatures of 253, 278 and 288 K) to fully compact the hydrate samples. 2) Wave speed increases at constant sample length strongly suggest grain to grain bonds form between adjacent ice or gas hydrate grains. The relative wave speed increases with time show this process is more efficient in ice samples, perhaps due to the higher mobility of water in ice's crystal lattice. 3) Within the pressure and temperature conditions studied, the wave speed based calculations of Poisson's ratio are 5 to 6\% smaller in sI and sII gas hydrate than in ice. 4) Shear wave speed decreases with increasing uniaxial pressure in Ice Ih, sI methane and sII methane-ethane hydrate. This unusual behavior was well known in ice Ih but was not known to apply to gas hydrates because no independent measurements of shear wave speed versus pressure in gas hydrates were previously available.