The stress singularity of the Phan-Thien-Tanner (PTT) fluid is determined for steady planar stick-slip and slip-stick flows. In the presence of a solvent viscosity, we show that the velocity field is Newtonian dominated local to the singularity. The velocity vanishes as v=O(r12), with r the radial distance from the singular point at the meeting of the solid (stick) and free (slip) surfaces. The solvent stresses thus behave as O(r-12). The polymer stresses are only slightly less singular O(r-411), but require boundary layers at both the stick and slip surfaces for their resolution. The stick surface boundary layer is of thickness O(r76) whilst the slip surface boundary layer is very slightly thicker O(r2320). Solutions are constructed for stick-slip and slip-stick flow regimes, both of which share the same asymptotic structure and singularity. The behaviour described here breaks down in the limit of vanishing solvent viscosity as well as the Oldroyd-B limit.