  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.
