We obtain the almost sure non-singularity of general Wigner ensembles of random matrices when the distribution of the entries are independent but not necessarily identically distributed and may depend on the size of the matrix. These models include adjacency matrices of random graphs and also sparse, generalized, universal and banded random matrices. We find universal rates of convergence and precise estimates for the probability of singularity which depend only on the size of the biggest jump of the distribution functions governing the entries of the matrix. Our proofs are based on a concentration function inequality due to Kesten and allows us to improve the known rates of convergence for the Wigner case when the distribution of the entries do not depend on the size of the matrix. This is joint work with Paulo Manrique and Victor Perez-Abreu..