We consider stagnation point flow away from a wall for creeping flow of dilute polymer solutions. For a simplified flow geometry, we explicitly show that a narrow region of strong polymer extension (a birefringent strand) forms downstream of the stagnation point in the UCM model and extensions, like the FENE-P model. These strands are associated with the existence of an essential singularity in the stresses, which is induced by the fact that the stagnation point makes the convective term in the constitutive equation into a singular point. We argue that the mechanism is quite general, so that all flows that have a separatrix going away from the stagnation point exhibit some singular behaviour. These findings are the counterpart for wall stagnation points of the recently discovered singular behaviour in purely elongational flows: the underlying mechanism is the same while the different nature of the singular stress behaviour reflects the different form of the velocity expansion close to the stagnation point. (C) 2008 Elsevier B.V. All rights reserved.