The quantum eigenstates and eigenvalues on a toroidal two dimensional phase-space are studied. To each eigenfunction is associated an integer, the Chern index, which tests the localization of the eigenfunction as some periodicity conditions are changed. The Chern index is a topological invariant which can only change when a spectral degeneracy occurs. These topological numbers are computed for three different models: two having an underlying regular dynamics, the third-one having a chaotic dynamics. The role played by the separatrix-states, the effects of quantum tunneling (symmetry effects) and of a classically chaotic dynamics in the spectrum of the Chern indices are discussed. The values taken by those indices are interpreted in terms of a phase-space distribution function. (author) 13 refs.; 12 figs.