When $X$ is a smooth projective variety, one can use Mori's bend-and-break trick to establish the cone theorem. However, when $X$ has singularity (say klt. singularity), the cone theorem is obtained by a series of hard results: vanishing theorem -> non-vanishing theorem -> rationality theorem -> cone theorem.
I was wondering what would go wrong when we follow the argument in the smooth case with some simple minded modifications --- like using resolution or cyclic covering? Moreover, the bend-and-break lemma itself does not require any smoothness.