An experimental campaign has been carried out with the aim of providing an insight into the fracture processes occurring during the cutting of different types of polymers, with features ranging from typically brittle to soft hyperelastic behaviour. The steady state of cutting is investigated using a sharp thin blade, and tracking the insertion force versus the penetration displacement. For soft, highly-deformable polymers, the influence of large deformations at the crack tip is also taken into account, through full-field finite strain maps obtained by means of digital image correlation. In brittle polymers, the influence of the cutting tool sharpness is discussed with respect to the onset of crack propagation, through a comparison with a simplified analytical model, and numerical finite element analyses. The results suggest that the propagation of the cut clearly depends on the tool sharpness, and for a class of brittle polymers it appears that this may happen as a stable fracture process, with the distance between the blade and the crack tip remaining constant during propagation. On the contrary, soft polymers appear to be much less sensible to the tool profile, and it is the large deformation that ultimately determines the fracture behaviour when the crack tip is reached by the blade.