  Let $\mathcal{G}_{n,r,s}$ denote a uniformly random $r$-regular $s$-uniform
hypergraph on the vertex set $\{1,2,\ldots, n\}$. We establish a threshold
result for the existence of a spanning tree in $\mathcal{G}_{n,r,s}$,
restricting to $n$ satisfying the necessary divisibility conditions.
Specifically, we show that when $s\geq 5$, there is a positive constant
$\rho(s)$ such that for any $r\geq 2$, the probability that
$\mathcal{G}_{n,r,s}$ contains a spanning tree tends to 1 if $r > \rho(s)$, and
otherwise this probability tends to zero. The threshold value $\rho(s)$ grows
exponentially with $s$. As $\mathcal{G}_{n,r,s}$ is connected with probability
which tends to 1, this implies that when $r \leq \rho(s)$, most $r$-regular
$s$-uniform hypergraphs are connected but have no spanning tree. When $s=3,4$
we prove that $\mathcal{G}_{n,r,s}$ contains a spanning tree with probability
which tends to 1, for any $r\geq 2$. Our proof also provides the asymptotic
distribution of the number of spanning trees in $\mathcal{G}_{n,r,s}$ for all
fixed integers $r,s\geq 2$. TPreviously, this asymptotic distribution was only
known in the trivial case of 2-regular graphs, or for cubic graphs.
