We evaluate regions of parameter space in the minimal supergravity model where ``unbounded from below'' (UFB) or charge or color breaking minima (CCB) occur. Our analysis includes the most important terms from the 1-loop effective potential. We note a peculiar discontinuity of results depending on how renormalization group improvement is performed: One case leads to a UFB potential throughout the model parameter space, while the other typically agrees quite well with similar calculations performed using only the tree level potential. We compare our results with constraints from cosmology and naturalness and find a preferred region of parameter space which implies $m_{\tg}\alt 725$ GeV, $m_{\tq}\alt 650$ GeV, $m_{\tw_1}\alt 225$ GeV and $m_{\tell_R}\alt 220$ GeV. We discuss the consequences of our results for supersymmetry searches at various colliding beam facilities.

22 pages plus 8 figures; uuencoded text file with PS figures also available via anonymous ftp from ftp://hep.fsu.edu/preprints/baer/vac.uu