The vibrations of a single-crystal germanium (Ge) membrane are studied in air and vacuum using laser vibrometry, in order to determine mechanical properties such as Q-factors, tensile stress, anisotropy, and robustness to shock. Resonance modes up to 3:2 are identified, giving a residual stress measurement of 0.22 GPa, consistent with the value obtained from x-ray relaxation studies. The membrane is found to be isotropic, with Q-factors ranging from around 40 at atmospheric pressure to over 3200 at mbar, significantly lower than those found in polycrystalline Ge micromechanical devices. The robustness to shock is explained through the high resonance mode frequencies and the dissipation mechanism into the substrate, which is a direct consequence of having a high quality film with low residual tensile stress, giving the potential for such films to be used in optoelectronic devices.