  Motivated by the possibilities of $b-\tau$ or $t-b-\tau$ Yukawa unification
in the supersymmetric Grand Unified Theories, we consider the dangerous
directions of the supersymmetric potential for large values of $\tan\beta (\geq
30)$, in two versions of the minimal supergravity model with and without common
soft breaking scalar masses at the GUT scale, where the potential may become
unbounded from below. We find that for the common trilinear coupling $A_0 \leq
0$ the requirement of $b-\tau$ unification in conjunction with the stability
condition on the potential yields highly restrictive sparticle spectra with
upper, and in many cases, lower bounds stronger than the available experimental
lower bounds, on the soft SUSY breaking common scalar mass and the common
gaugino mass. Over a significant region of the parameter space, the model
becomes even more restrictive if the common sfermion soft mass is different
from the soft mass for the Higgs sector. We also find that the bulk of this
restricted parameter space can be probed at the LHC. Im models with $t-b-\tau$
Yukawa unification, $A_0 \leq 0$ is ruled out from potential constraints.
