  We consider a version of the Lipman-Zariski conjecture for logarithmic vector
fields and logarithmic $1$-forms on pairs. Let $(X,D)$ be a pair consisting of
a normal complex variety $X$ and an effective Weil divisor $D$ such that the
sheaf of logarithmic vector fields (or dually the sheaf of reflexive
logarithmic $1$-forms) is locally free. We prove that in this case the
following holds: If $(X,D)$ is dlt, then $X$ is necessarily smooth and $\lfloor
D\rfloor $ is snc. If $(X,D)$ is lc or the logarithmic $1$-forms are locally
generated by closed forms, then $(X,\lfloor D\rfloor)$ is toroidal.
