  A 7-dimensional area-minimizing embedded hypersurface $M$ will in general
have a discrete singular set. The same is true if $M$ is stable, or has bounded
index, provided $H^6(sing M) = 0$. We show that if $M_i$ are a sequence of such
minimal hypersurfaces which are minimizing, stable, or have bounded index, then
$M_i$ can limit to a singular $M$ with only very controlled geometry, topology,
and singular set. We show one can always "parameterize" a subsequence $i'$ with
controlled bi-Lipschitz maps $\phi_{i'}$ taking $\phi_{i'}(M_{1'}) = M_{i'}$.
As a consequence, we prove the space of smooth, closed, embedded minimal
hypersurfaces $M$ in a closed Riemannian 8-manifold $(N, g)$ with a priori
bounds $H^7(M) \leq \Lambda$ and $index(M) \leq I$ divides into finitely-many
diffeomorphism types, and this finiteness continues to hold (in a suitable
sense) if one allows the metric g to vary, or M to be singular.
