Observations are presented of low-frequency magnetic oscillations in the Martian magnetosheath, magnetic pileup region, and tail as observed by the Magnetometer/Electron Reflectometer experiment on board Mars Global Surveyor. Within the dayside magnetosheath the oscillations are found to be predominantly compressional, elliptically polarized waves with wave vectors that have large angles relative to the mean field and dominant frequencies that are significantly below the local proton gyrofrequency. On the basis of these observations we identify these oscillations as mirror mode instabilities. In the nightside magnetosheath the oscillations are predominantly transverse and elliptical, and they propagate at smaller angles relative to the mean field at frequencies within a factor of 2--10 less than the local proton gyrofrequency. These characteristics lead us to associate these oscillations with ion/ion-resonant instabilities that arise from counterstreaming plasma populations such as the solar wind and pickup ions of planetary origin. Within the magnetic pileup region and tail the oscillations have considerably smaller amplitudes and are linearly polarized, obliquely propagating, ultralow-frequency oscillations which may be a mix of multiple wave modes.