In this paper, we consider generic corank 2 sub-Riemannian structures (i.e., nonlinear optimal control problems that are linear in the control and have quadratic cost), and we show that the spherical Hausdorff measure is always a C-1-smooth volume, which is in fact generically C-2-smooth out of a stratified subset of codimension 7. In particular, for rank 4, it is generically C-2. This result is fundamental to study the diffusion of the heat and other evolution phenomena in sub-Riemannian geometry.