  We study the distribution of the least singular value associated to an
ensemble of sparse random matrices. Our motivating example is the ensemble of
$N\times N$ matrices whose entries are chosen independently from a Bernoulli
distribution with parameter $p$. These matrices represent the adjacency
matrices of random Erd\H{o}s--R\'enyi digraphs and are sparse when $p\ll 1$. We
prove that in the regime $pN\gg 1$, the distribution of the least singular
value is universal in the sense that it is independent of $p$ and equal to the
distribution of the least singular value of a Gaussian matrix ensemble. We also
prove the universality of the joint distribution of multiple small singular
values. Our methods extend to matrix ensembles whose entries are chosen from
arbitrary distributions that may be correlated, complex valued, and have
unequal variances.
