  We study the regularity of the free boundary in the obstacle for the
$p$-Laplacian, $\min\bigl\{-\Delta_p u,\,u-\varphi\bigr\}=0$ in
$\Omega\subset\mathbb R^n$. Here, $\Delta_p u=\textrm{div}\bigl(|\nabla
u|^{p-2}\nabla u\bigr)$, and $p\in(1,2)\cup(2,\infty)$.
  Near those free boundary points where $\nabla \varphi\neq0$, the operator
$\Delta_p$ is uniformly elliptic and smooth, and hence the free boundary is
well understood. However, when $\nabla \varphi=0$ then $\Delta_p$ is singular
or degenerate, and nothing was known about the regularity of the free boundary
at those points.
  Here we study the regularity of the free boundary where $\nabla \varphi=0$.
On the one hand, for every $p\neq2$ we construct explicit global
$2$-homogeneous solutions to the $p$-Laplacian obstacle problem whose free
boundaries have a corner at the origin. In particular, we show that the free
boundary is in general not $C^1$ at points where $\nabla \varphi=0$. On the
other hand, under the "concavity" assumption $|\nabla \varphi|^{2-p}\Delta_p
\varphi<0$, we show the free boundary is countably $(n-1)$-rectifiable and we
prove a nondegeneracy property for $u$ at all free boundary points.
