  In the classical obstacle problem, the free boundary can be decomposed into
"regular" and "singular" points. As shown by Caffarelli in his seminal papers
\cite{C77,C98}, regular points consist of smooth hypersurfaces, while singular
points are contained in a stratified union of $C^1$ manifolds of varying
dimension. In two dimensions, this $C^1$ result has been improved to
$C^{1,\alpha}$ by Weiss \cite{W99}.
  In this paper we prove that, for $n=2$ singular points are locally contained
in a $C^2$ curve. In higher dimension $n\ge 3$, we show that the same result
holds with $C^{1,1}$ manifolds (or with countably many $C^2$ manifolds), up to
the presence of some "anomalous" points of higher codimension. In addition, we
prove that the higher dimensional stratum is always contained in a
$C^{1,\alpha}$ manifold, thus extending to every dimension the result in
\cite{W99}.
  We note that, in terms of density decay estimates for the contact set, our
result is optimal. In addition, for $n\ge3$ we construct examples of very
symmetric solutions exhibiting linear spaces of anomalous points, proving that
our bound on their Hausdorff dimension is sharp.
