We consider quotients of spheres by linear actions of real tori and finite abelian groups. To each quotient we associate a matroid or sequence of matroids. In the case of real tori, we find the integral homology groups of the resulting quotient spaces and singular sets in terms of the Tutte polynomial of the matroid(s). For finite groups, an algorithm for computing the Zp -homology of the quotient space is given.