  We prove the Lipman-Zariski conjecture for complex surface singularities of
genus one, and also for those of genus two whose link is not a rational
homology sphere. As an application, we characterize complex $2$-tori as the
only normal compact complex surfaces whose smooth locus has trivial tangent
bundle. We also deduce that all complex-projective surfaces with locally free
and generically nef tangent sheaf are smooth, and we classify them.
