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Recent observations have revealed a trove of unexpected morphological features in many of the Milky Way's stellar streams. Explanations for such features include time-dependent deformations of the Galactic gravitational potential, local disruptions induced by dark matter substructure, and special configurations of the streams' progenitors. In this paper, we study how these morphologies can also arise in certain static, non-spherical gravitational potentials that host a subset of resonantly-trapped orbit families. The transitions, or separatrices, between these orbit families mark abrupt discontinuities in the orbital structure of the potential. We develop a novel numerical approach for measuring the libration frequencies of resonant and near-resonant orbits, and apply it to study the evolution of stellar streams on these orbits. We reveal two distinct morphological features that arise in streams on near-resonant orbits: fans, that come about due to a large spread in the libration frequencies near a separatrix; and bifurcations, that arise when a separatrix splits the orbital distribution of the stellar stream between two (or more) distinct orbit families. We demonstrate that these effects can arise in some Milky Way streams for certain choices of the dark matter halo potential, and discuss how this might be used to probe and constrain the global shape of the Milky Way's gravitational potential.