  We study mapping class group orbits of homotopy and isotopy classes of curves
with self-intersections. We exhibit the asymptotics of the number of such
orbits of curves with a bounded number of self-intersections, as the complexity
of the surface tends to infinity. We also consider the minimal genus of a
subsurface that contains the curve. We determine the asymptotic number of
orbits of curves with a fixed minimal genus and a bounded self-intersection
number, as the complexity of the surface tends to infinity. As a corollary of
our methods, we obtain that most curves that are homotopic are also isotopic.
Furthermore, using a theorem by Basmajian, we get a bound on the number of
mapping class group orbits on a given a hyperbolic surface that can contain
short curves. For a fixed length, this bound is polynomial in the signature of
the surface. The arguments we use are based on counting embeddings of ribbon
graphs.
