Whether there is global regularity or finite time blow-up forthe space homogeneous Landau equation with Coulomb potential is a longstanding open problem in the mathematical analysis of kinetic models. Thistalk shows that the Hausdorff dimension of the set of singular times of theglobal weak solutions obtained by Villanis procedure is at most 1/2. (Workin collaboration with M.P. Gualdani, C. Imbert and A. Vasseur)