  We consider the transport equation on $[0,T]\times \mathbb{R}^n$ in the
situation where the vector field is $BV$ off a set $S\subset [0,T]\times
\mathbb{R}^n$. We demonstrate that solutions exist and are unique provided that
the set of singularities has a sufficiently small anisotropic fractal dimension
and the normal component of the vector field is sufficiently integrable near
the singularities. This result improves upon recent results of Ambrosio who
requires the vector field to be of bounded variation everywhere. In addition,
we demonstrate that under these conditions almost every trajectory of the
associated regular Lagrangian flow does not intersect the set $S$ of
singularities. Finally, we consider the particular case of an initial set of
singularities that evolve in time so the singularities consists of curves in
the phase space, which is typical in applications such as vortex dynamics. We
demonstrate that solutions of the transport equation exist and are unique
provided that the box-counting dimension of the singularities is bounded in
terms of the H\"older exponent of the curves.
