We present an experimental investigation of instabilities due to the formation of interfacial micellar material around a flowing liquid column. By injecting an aqueous surfactant solution (cetylpyridinium chloride) into a reservoir of organic oil (oleic acid), we observe a wide variety of morphological states, dependent on the imposed flow rate and surfactant concentration. Interfacial rheology measurements show that this material is viscoelastic. The states observed include connected and disconnected falling droplets, a moving pipelike column with various surface wave instabilities, and a straight cylindrical pipe which undergoes a buckling instability. These states result from the competition between surface tension type effects, the geometry of the flow, and the growth of the micellar material at the interface.