  We prove that the singular locus of the commuting variety of a noncommutative
reductive Lie algebra is contained in the irregular locus and we compute the
codimension of the latter. We prove that one of the irreducible components of
the irregular locus has codimension 4. This yields the lower bound of the
codimension of the singular locus, in particular, implies that it is at least
2. We also prove that the commuting variety is rational.
