We present new results on the nonlinear stability of Bingham fluid Poiseuille flows in pipes and plane channels. These results show that the critical Reynolds number for transition, Rec, increases with Bingham number, B, at least as fast as Re-c similar to B-1/2 as B --> infinity. Estimates for the rate of increase are also provided. We compare these bounds and existing linear stability bounds with predictions from a series of phenomenological criteria for transition, as B --> infinity, concluding that only Hanks [AIChE J. 9 (1963) 306; 15 (1) (1963) 25] criteria can possibly be compatible with the theoretical criteria as B --> infinity. In the more practical range of application, 0 less than or equal to B less than or equal to 50, we show that there exists a large disparity between the different phenomenological criteria that have been proposed.