Building on results of an earlier paper we study the discrete spectrum of the Rayleigh piston. We first prove the absence of discrete spectrum on the subspace of odd functions everywhere in the Lorentz regime. Then we give upper bounds on the number of discrete eigenvectors as a function of the mass ratio using a variety of methods which to some degree complement each other. We also investigate the precise degree of divergence of these bounds as the mass ratio goes to infinity respectively zero.