We study algorithms for estimating the statistical leverage scores of rectangular dense or sparse matrices of arbitrary rank. Our approach is based on combining rank revealing methods with compositions of dense and sparse randomized dimensionality reduction transforms. We first develop a set of fast novel algorithms for rank estimation, column subset selection, and least squares preconditioning. We then describe the design and implementation of leverage score estimators based on these primitives. These estimators are also effective for rank deficient input, which is frequently the case in data analytics applications. We provide detailed complexity analyses for all algorithms as well as meaningful approximation bounds and comparisons with the state of the art. We conduct extensive numerical experiments to evaluate our algorithms and to illustrate their properties and performance using synthetic and real world datasets.