  We show that for a generic $8$-dimensional Riemannian manifold with positive
Ricci curvature, there exists a smooth minimal hypersurface. Without the
curvature condition, we show that for a dense set of 8-dimensional Riemannian
metrics there exists a minimal hypersurface with at most one singular point.
This extends previous work on generic regularity that only dealt with
area-minimizing hypersurfaces. These results are a consequence of a more
general estimate for a one-parameter min-max minimal hypersurface $\Sigma
\subset (M,g)$ (valid in any dimension): $$\mathcal H^{0}
(\mathcal{S}_{nm}(\Sigma)) +{\rm Index}(\Sigma) \leq 1$$ where
$\mathcal{S}_{nm}(\Sigma)$ denotes the set of singular points of $\Sigma$ with
a unique tangent cone non-area minimizing on either side.
