  The tangential ramification locus $B_{X/Y}^t\subset B_{X/Y}$ is the subset of
points in the ramification locus where the sheaf of relative vector fields
$T_{X/Y}$ fails to be locally free. It was conjectured by Zariski and Lipman
that if $V/k$ is a variety over a field $k$ of characteristic 0 and $B^t_{V/k}=
\emptyset$, then $V/k$ is smooth (=regular). We prove this conjecture when
$V/k$ is a locally complete intersection. We prove also that $B_{V/k}^t=
\emptyset$ implies $\operatorname {codim}_V B_{V/k}\leq 1 $ in positive
characteristic, if $V/k $ is the fibre of a flat morphism satisfying generic
smoothness.
