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In this talk, I will systematically approach the study of threshold singularities of Feynman diagrams in momentum space, using methods from convex geometry and graph theory. I will also describe how non-pinched threshold singularities can be regularised by analytically continuing the integrand, provided monodromy constraints are satisfied. I will conclude by discussing the cancellation of collinear and soft final-state singularities in differential cross-sections through the Local Unitarity formalism, and briefly comment on the repercussions of extending such cancellation mechanism to initial-state singularities.