  We investigate the family of electrostatic spherically symmetric solutions of
the five-dimensional Kaluza-Klein theory. Both charged and neutral cases are
considered. The analysis of the solutions, through their geometrical
properties, reveals the existence of black holes, wormholes and naked
singularities. A new class of regular solutions is identified. A monopole
perturbation study of all these solutions is carried out, enabling us to prove
analytically the stability of large classes of solutions. In particular, the
black hole solutions are stable, while for the regular solutions the stability
analysis leads to an eigenvalue problem.
