Viscosity regularisation methods are probably the most popular current method for computing visco-plastic fluid flows. They are however generally used in an ad hoc manner. Here we examine convergence of regularised solutions to those of the corresponding exact models, in both mathematical and physical senses. Mathematically, the aim is to give practical guidance as to the order of error that one might expect for different regularisations and for different types of flow. Our theoretical results are illustrated with a number of computed example flows showing the orders of error predicted. Physically, the question is whether or not the regularised solutions behave in the same way as the exact solutions, qualitatively as well as quantitatively. We show that there are flows for which regularisation methods will generate their maximum errors, e.g. lubrication-type flows. In this context, we also consider the effects of regularisation on problems of hydrodynamic stability. For broad classes of problems, stability characteristics of the flow are incorrectly predicted by the use of viscosity regularisation methods. © 2005 Elsevier B.V. All rights reserved.