For a discrete group with smooth classifying space acting smoothly on a smooth manifold, we use Kasparov theory and the Baum-Connes apparatus to analyze various kinds of spectral triples and Fredholm modules one can associate to this situation, from a topological point of view. The main unifying definition we introduce is of the `Dirac class' of such a smooth action, and the homological properties of Dirac classes, established using the theory of topological correspondences,are discussed, as well as some secondary invariants of them.