We demonstrate shear stress sensing with a Bragg grating-based microstructured optical fiber sensor embedded in a single lap adhesive joint. We achieved an unprecedented shear stress sensitivity of 59.8 pm/MPa when the joint is loaded in tension. This corresponds to a shear strain sensitivity of 0.01 pm/µε. We verified these results with 2D and 3D finite element modeling. A comparative FEM study with conventional highly birefringent side-hole and bow-tie fibers shows that our dedicated fiber design yields a fourfold sensitivity improvement.