Sensor selection is an NP-hard problem involving the selection of S out of N sensors such that optimal filtering performance is attained. We present a novel approach for sensor selection that utilizes a heuristic measure quantifying the incoherence of the vector space spanned by the sensors with respect to the system's principal directions. This approach facilitates the formulation of a convex relaxation problem that can be efficiently modeled and solved using compressed sensing (CS) algorithms. We subsequently develop a new CS algorithm based on subgradient projections. The new CS algorithm for sensor selection is shown to outperform existing methods in a number of applications.