  A mean curvature flow starting from a closed embedded hypersurface in
$R^{n+1}$ must develop singularities. We show that if the flow has only generic
singularities, then the space-time singular set is contained in finitely many
compact embedded $(n-1)$-dimensional Lipschitz submanifolds plus a set of
dimension at most $n-2$. If the initial hypersurface is mean convex, then all
singularities are generic and the results apply.
  In $R^3$ and $R^4$, we show that for almost all times the evolving
hypersurface is completely smooth and any connected component of the singular
set is entirely contained in a time-slice. For $2$ or $3$-convex hypersurfaces
in all dimensions, the same arguments lead to the same conclusion: the flow is
completely smooth at almost all times and connected components of the singular
set are contained in time-slices. A key technical point is a strong
{\emph{parabolic}} Reifenberg property that we show in all dimensions and for
all flows with only generic singularities. We also show that the entire flow
clears out very rapidly after a generic singularity.
  These results are essentially optimal.
