  Let $d\geq 3$ be a fixed integer, and a prime number $p$ such that
$\gcd(p,d)=1$. Let $A$ be the adjacency matrix of a random $d$-regular directed
graph on $n$ vertices. We show that as a random matrix in ${\mathbb F}_p$,
\begin{equation} {\mathbb P}(\text{$A$ is singular in ${\mathbb F}_p$})\leq
\frac{1+{\mathrm{o}}(1)}{p-1}, \end{equation} as $n$ goes to infinity. As a
consequence, as a random matrix in $\mathbb R$, \begin{equation} {\mathbb
P}(\text{$A$ is singular in $\mathbb R$})={\mathrm{o}}(1) \end{equation} as $n$
goes to infinity. This answers an open problem by Frieze [12] and Vu [29,30],
for random $d$-regular bipartite graphs. The proof combines a local central
limit theorem and a large deviation estimate.
