We consider the combinatorial properties of the trace of a random walk on the
complete graph and on the random graph $G(n,p)$. In particular, we study the
appearance of a fixed subgraph in the trace. We prove that for a subgraph
containing a cycle, the threshold for its appearance in the trace of a random
walk of length $m$ is essentially equal to the threshold for its appearance in
the random graph drawn from $G(n,m)$. In the case where the base graph is the
complete graph, we show that a fixed forest appears in the trace typically much
earlier than it appears in $G(n,m)$.