One of the central achievements in combinatorial random matrix theory is the understanding of the singularity probability of a random n x n matrix with independent {1,-1}-Bernoulli entries, where each entry is 1 with some probability p and -1 with probability 1-p. In particular it has been established that the singularity probability exhibits a phase transition if p is of order log(n)/n. In this project, we aim to establish similar phase transition results for more complicated random matrix models, such as random matrices with constraints, and we will study such phase transitions for more general polynomial anticoncentration problems.