  We investigate the regularity of the free boundary for the Signorini problem
in $\mathbb{R}^{n+1}$. It is known that regular points are $(n-1)$-dimensional
and $C^\infty$. However, even for $C^\infty$ obstacles $\varphi$, the set of
non-regular (or degenerate) points could be very large, e.g. with infinite
$\mathcal{H}^{n-1}$ measure.
  The only two assumptions under which a nice structure result for degenerate
points has been established are: when $\varphi$ is analytic, and when
$\Delta\varphi < 0$. However, even in these cases, the set of degenerate points
is in general $(n-1)$-dimensional (as large as the set of regular points).
  In this work, we show for the first time that, "usually", the set of
degenerate points is small. Namely, we prove that, given any $C^\infty$
obstacle, for "almost every" solution the non-regular part of the free boundary
is at most $(n-2)$-dimensional. This is the first result in this direction for
the Signorini problem.
  Furthermore, we prove analogous results for the obstacle problem for the
fractional Laplacian $(-\Delta)^s$, and for the parabolic Signorini problem. In
the parabolic Signorini problem, our main result establishes that the
non-regular part of the free boundary is $(n-1-\alpha_\circ)$-dimensional for
almost all times $t$, for some $\alpha_\circ > 0$.
  Finally, we construct some new examples of free boundaries with degenerate
points.
