We study the gaps between consecutive singular values of random rectangular matrices. Specifically, if $M$ is an $n \times p$ random matrix with independent and identically distributed entries and $Σ$ is a $n \times n$ deterministic positive definite matrix, then under some technical assumptions we give lower bounds for the gaps between consecutive singular values of $Σ^{1/2} M$. As a consequence, we show that sample covariance matrices have simple spectrum with high probability. Our results resolve a conjecture of Vu [{\em Probab. Surv.}, 18:179--200, 2021]. We also discuss some applications, including a bound on the spacings of eigenvalues of the adjacency matrix of random bipartite graphs.