The stability of the moving contact line is examined for both Newtonian and viscoelastic fluids. Two methods for relieving the contact line singularity are chosen: matching the free surface profile to a precursor film of thicknessb, and introducing slip at the solid substrate. The linear stability of the Newtonian capillary ridge with the precursor film model was first examined by Troianetal. [Europhys. Lett.10, 25 (1989)]. Using energy analysis, we show that in this case the stability of the advancing capillary ridge is governed by rearrangement of fluid in the flow direction, whereby thicker regions develop that advance more rapidly under the influence of a body force. In addition, we solve the Newtonian linear stability problem for the slip model and obtain results very similar to those from the precursor film model. Interestingly, stability results for the two models comparequantitativelywhen the precursor film thicknessbis numerically equal to the slip parameter &agr;. With the slip model, it is possible to examine the effect of contact angle on the stability of the advancing front, which, for small contact angles, was found to be independent of the contact angle. The stability of an Oldroyd‐B fluid was examined via perturbation theory in Weissenberg number. It is found that elastic effects tend to stabilize the capillary ridge for the precursor film model, and this effect is more pronounced as the precursor film thickness is reduced. The perturbation result was examined in detail, indicating that viscoelastic stabilization arises primarily due to changes of momentum transfer in the flow direction, while elasticity has little effect on the response of the fluid to flow in the spanwise direction. ©1996 American Institute of Physics.