It is shown that there are many stable configurations for a domain wall pinned by a notch along a magnetic stripe. The stability of several of these configurations is investigated numerically as a function of the thickness of the magnetic film. The depinning mechanism depends on the structure of the domain wall and on the thickness of the magnetic film. In the case of a spin-valve structure, it appears that the stray fields emerging from the hard layer at the notch location influence the stability of the micromagnetic configuration. Different depinning mechanisms are thus observed for the same film thickness depending on the magnetization orientation of the propagating domain. This conclusion qualitatively explains experimental magnetoresistance measurements.