  A celebrated result of Mirzakhani states that, if $(S,m)$ is a finite area
\emph{orientable} hyperbolic surface, then the number of simple closed
geodesics of length less than $L$ on $(S,m)$ is asymptotically equivalent to a
positive constant times $L^{\dim\mathcal{ML}(S)}$, where $\mathcal{ML}(S)$
denotes the space of measured laminations on $S$. We observed on some explicit
examples that this result does not hold for \emph{nonorientable} hyperbolic
surfaces. The aim of this article is to explain this surprising phenomenon. Let
$(S,m)$ be a finite area \emph{nonorientable} hyperbolic surface. We show that
the set of measured laminations with a closed one--sided leaf has a peculiar
structure. As a consequence, the action of the mapping class group on the
projective space of measured laminations is not minimal. We determine a partial
classification of its orbit closures, and we deduce that the number of simple
closed geodesics of length less than $L$ on $(S,m)$ is negligible compared to
$L^{\dim\mathcal{ML}(S)}$. We extend this result to general multicurves. Then
we focus on the geometry of the moduli space. We prove that its Teichm\"uller
volume is infinite, and that the Teichm\"uller flow is not ergodic. We also
consider a volume form introduced by Norbury. We show that it is the right
generalization of the Weil--Petersson volume form. The volume of the moduli
space with respect to this volume form is again infinite (as shown by Norbury),
but the subset of hyperbolic surfaces whose one--sided geodesics have length at
least $\varepsilon>0$ has finite volume. These results suggest that the moduli
space of a nonorientable surface looks like an infinite volume geometrically
finite orbifold. We discuss this analogy and formulate some conjectures.
