  Let $d \geq 3$ be a fixed integer. We give an asympotic formula for the
expected number of spanning trees in a uniformly random $d$-regular graph with
$n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if
$d$ is odd.) We also obtain the asymptotic distribution of the number of
spanning trees in a uniformly random cubic graph, and conjecture that the
corresponding result holds for arbitrary (fixed) $d$. Numerical evidence is
presented which supports our conjecture.
