Sharp singularity asymptotics for discrete random matrices

We resolve the Rademacher singularity conjecture: an $n\times n$ matrix with independent uniform $\{-1,1\}$ entries is singular with probability $(2+o(1))n^22^{-n}$. More generally, for an $n\times n$ matrix $M_n$ with independent entries uniform on a fixed set $S\subset\mathbb{R}$ of cardinality