Yes, graphs come up in various moduli contexts quite a bit! As the previous two answers indicated, $Out(F_n)$ is closely related to graphs. This goes back to the original paper of Vogtmann and Culler where outer space was first introduced. $BOut(F_n)$ can be modelled as the moduli space of metric graphs with first Betti number equal to $n$. If you consider the moduli space of graphs as an orbifold then this has the right integral homotopy type (but if you take the coarse quotient space then it is just a rational classifying space because $Out(F_n)$ acts on outer space with finite stabilisers).
Given a cyclic operad $P$ in the category of topological spaces one can talk about the space of graphs with vertices labelled by $P$. An ordinary graph is the same as a $Comm$-labelled graph since the commutative cyclic operad is just a point in each arity. A ribbon graph is the same as an $Assoc$-labelled graph since in cyclic arity $n$ the associative cyclic operad is the set of cyclic oderings on $n$ letters.
So the Culler-Vogtmann work can be interpreted as saying that $BOut(F_n)$ is the moduli space of rank $n$ $Comm$-graphs. Moduli spaces of Riemann surfaces with marked points are are homotopy equivalent to spaces of $Assoc$-graphs. There are some other results of this type. A Mobius graph is like a ribbon graph but with edges possibly given a half-twist; these are the same as graphs labelled by the cyclic operad $InvAss$ for associative algebras with an involution (perhaps could be called the hermitian associative operad). The spaces of Mobius graphs are homotopy equivalent to the moduli spaces of surfaces with a Klein structure (an unoriented version of complex structure), or equivalently, the classifying spaces of the mapping class groups of unorientable surfaces. Another result of this type is that the space of graphs labelled by the framed little 2-discs cyclic operad is homotopy equivalent to the moduli space of 3-dimensional oriented handlebodies.
To expand on some of Jim's comments a bit further: The connection to graph homology is as follows. First you need to know that there is a duality functor for cyclic operads in chain complexes. Sometimes it is called dg-duality, and sometimes it is called the Bar construction or Koszul duality. (Strictly speaking, the duality is given by a bar construction that produces a cyclic cooperad, followed by applying linear duality to turn in back into a cyclic operad; this construction agrees up to quasi-isomorphism with the koszul duality construction for cyclic operads that are Koszul.) The associative cyclic operad is self-dual. The commutative cyclic operad is dual to the Lie cyclic operad.
The general theorem is that if $P$ is a cyclic operad in topological spaces, then $C_*P$ is a cyclic operad in chain complexes with dual $D(C_*P)$, and $D(C_*P)$-graph homology computes the cohomology of the space of $P$-labelled graphs. This is why $Lie$ graph homology computes the cohomology of $Out(F_n)$ and $Ass$ graph homology computes the cohomology of moduli spaces of Riemann surfaces.