Singularity of discrete random matrices

Let $M_n$ be an $n \times n$ random matrix whose entries are i.i.d. copies of a discrete random variable $\xi$. It has been conjectured that the dominant reason for the singularity of $M_n$ is the event that a row or column of $M_n$ is zero, or that two rows or columns of $M_n$ coincide (up to a sign). I will discuss joint work with Ashwin Sah (MIT) and Mehtaab Sawhney (MIT), towards the resolution of this conjecture.