An instability mechanism is shown to operate in complex, non-Newtonian fluids in which a jump in normal stresses occurs between two fluids. Fluids with a negative second normal stress difference can be unstable with respect to transverse or spanwise perturbations. The mechanism appears to be generic, although the details will depend on the specific flow and the nature of the complex fluid. The instability mechanism is illustrated for two-layer Couette and falling film flows of viscous suspensions.