  Let $\xi$ be a non-constant real-valued random variable with finite support,
and let $M_{n}(\xi)$ denote an $n\times n$ random matrix with entries that are
independent copies of $\xi$. For $\xi$ which is not uniform on its support, we
show that \begin{align*} \mathbb{P}[M_{n}(\xi)\text{ is singular}] &=
\mathbb{P}[\text{zero row or column}] + (1+o_n(1))\mathbb{P}[\text{two equal
(up to sign) rows or columns}], \end{align*} thereby confirming a folklore
conjecture. As special cases, we obtain:
  (1) For $\xi = \text{Bernoulli}(p)$ with fixed $p \in (0,1/2)$,
\[\mathbb{P}[M_{n}(\xi)\text{ is singular}] = 2n(1-p)^{n} +
(1+o_n(1))n(n-1)(p^2 + (1-p)^2)^{n},\] which determines the singularity
probability to two asymptotic terms. Previously, no result of such precision
was available in the study of the singularity of random matrices.
  (2) For $\xi = \text{Bernoulli}(p)$ with fixed $p \in (1/2,1)$,
\[\mathbb{P}[M_{n}(\xi)\text{ is singular}] = (1+o_n(1))n(n-1)(p^2 +
(1-p)^2)^{n}.\] Previously, only the much weaker upper bound of $(\sqrt{p} +
o_n(1))^{n}$ was known due to the work of Bourgain-Vu-Wood.
  For $\xi$ which is uniform on its support:
  (1) We show that \begin{align*} \mathbb{P}[M_{n}(\xi)\text{ is singular}] &=
(1+o_n(1))^{n}\mathbb{P}[\text{two rows or columns are equal}]. \end{align*}
(2) Perhaps more importantly, we provide a sharp analysis of the contribution
of the `compressible' part of the unit sphere to the lower tail of the smallest
singular value of $M_{n}(\xi)$.
