  Locally stable minimal hypersurface could have singularities in dimension
$\geq 7$ in general, locally modeled on stable and area-minimizing cones in the
Euclidean spaces. In this paper, we present different aspects of how these
singularities may affect the local behavior of minimal hypersurfaces.
  First, given a non-degenerate minimal hypersurface with strictly stable and
strictly minimizing tangent cone at each singular point, under any small
perturbation of the metric, we show the existence of a nearby minimal
hypersurface under new metric. For a generic choice of perturbation direction,
we show the entire smoothness of the resulting minimal hypersurface.
  Second, given a strictly stable minimal hypersurface $\Sigma$ with strictly
minimizing tangent cone at each singular point, we show that $\Sigma$ is
uniquely homologically minimizing in its neighborhood.
  Lastly, given a family of stable minimal hypersurfaces multiplicity $1$
converges to a stable minimal hypersurface $\Sigma$ in an eight manifold, we
show that there exists a non-trivial Jacobi field on $\Sigma$ induced by these
converging hypersurfaces, which generalizes a result by Ben Sharp in smooth
case; Moreover, positivity of this Jacobi field implies smoothness of the
converging sequence.
