We give a systematic study of certain class of generic Banach spaces. We show that they distinguish between an array of different properties related to smoothness of equivalent norms such as for example the Mazur intersection property or the existence of convex sets supported by all of their points. We also examine the dual constructions of generic Choquet simplexes with extra requirements such as for example those of Poulsen and Bauer asking that the set of extremal points is dense or closed, respectively.
