We developed a unified approach to study the largest and smallest singular values of sparse rectangular random matrices, both above the critical log(n)/n sparsity regime and at criticality. The approach is based on the non-backtracking operator and the Ihara-Bass formula for general Hermitian matrices with bipartite block structure, as well as on a graph-based approximation scheme developed by Alt, Ducatez, and Knowles (2019). Above criticality, the bounds are given in terms of the maximum and, respectively, minimum ℓ2 norms of the rows and columns of the variance profile matrix, and they work for the inhomogeneous case. At criticality, we study the centered adjacency matrices of homogeneous Erd˝os–R´enyi random graphs and find outlier thresholds depending on the aspect ratio. This is joint work with Yizhe Zhu, Haixiao Wang, and Zhichao Wang.