  We study the singular part of the free boundary in the obstacle problem for
the fractional Laplacian, \ $\min\bigl\{(-\Delta)^su,\,u-\varphi\bigr\}=0$ in
$\mathbb R^n$, for general obstacles $\varphi$. Our main result establishes the
complete structure and regularity of the singular set. To prove it, we
construct new monotonicity formulas of Monneau-type that extend those in
\cite{GP} to all $s\in(0,1)$.
