In this talk we survey some interesting problems related to the singularity problem of random discrete matrices. In particular, we discuss the following problems:1. an extension of an old problem of Odlyzko about the probability for randomly chosen $\pm 1$ vectors to intersect the hypercube \{1,-1\}^n in a non-trivial way, and 
2. an approximate version of a problem of Vu about the probability that a random matrix can be made singular after changing `not too many' entries by an adversary. 
 
This is based on joint works with Afonso Bandeira and Matthew Kwan, and on a recent work with Kyle Luh, Gweneth McKinley and Wojtek Samotij.