The main aspect of the underlying graph of a random $m$-out graph (each vertex is given a random set of $m$ out-edges) that has been studied is the fact that they are a.s. Hamiltonian.

Denoting this graph by $D_m(n)$, it is known that $D_2$ almost surely contains a vertex adjacent to three vertices of degree two, and in the paper <a href="http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V00-46D1P4C-3B&_user=1010281&_coverDate=12%2F31%2F1983&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_rerunOrigin=scholar.google&_acct=C000050264&_version=1&_urlVersion=0&_userid=1010281&md5=09e6546719ef9ff884b3ea6d5495466f&searchtype=a">"On the existence of Hamiltonian cycles in a class of random graphs"</a> Fenner and Frieze proved that $D_{23}$ is a.s. Hamiltonian. Finally Bohman and Frieze recently proved that $D_3$ is almost surely Hamiltonian in <a href="http://arxiv.org/abs/0904.0431?context=math">"Hamiltonian cycles in $3$-out"</a>.