The local asymptotic behaviour at the stick-slip singularity is determined for the Giesekus fluid in the presence of a solvent viscosity. In planar steady flow, the method of matched asymptotic expansions is used to show that it comprises a three region structure. Specifically, an outer or core region that links boundary layers at the rigid stick and free slip surfaces. In the outer region, the velocity field is shown to be Newtonian at leading order, with solvent stresses dominating the polymer stresses. In terms of the radial distance r from the singularity at the join of the stick and slip surfaces, the velocity field vanishes as O(r(1/2)). Consequently, the singular velocity gradients and solvent stresses are of O(r(-1/2)) with the less singular polymer stresses being shown to be O(r(-5/16)). The solvent and polymer stresses become comparable near the rigid stick and free slip surfaces, where boundary layers are required. These are of thickness O(r(5/4)) at the rigid stick surface and thickness O(r(17/14)) at the free slip surface. Solutions are constructed for both stick-slip and slip-stick flow regimes. These asymptotic results do not hold for the Oldroyd-B model nor for the case when the solvent viscosity is absent. (C) 2014 Elsevier B.V. All rights reserved.