Abstract This paper concerns the inverse mean curvature flow (IMCF) running from the boundary of a convex body that has no regularity assumption. We study the evolution of singularities by looking at the blow-up tangent cone around each singular point. We prove that the cone also evolves by the IMCF and that each singularity is removed when the evolving cone becomes flat. As a result, we derive the exact waiting time for a weak solution to become a smooth solution. In particular, necessary and sufficient condition for the existence of a smooth classical solution is given.