  The description of point defects in chiral liquid crystals via topological
methods requires the introduction of singular contact structures, a
generalisation of regular contact structures where the plane field may have
singularities at isolated points. We characterise the class of singularities
that may arise in such structures, as well as the subclass of singularities
that can occur in a Beltrami field. We discuss questions of global existence,
and prove that all singular contact structures with nonremovable singularities
are overtwisted. To connect the theory to experiment we also discuss normal and
tangential boundary conditions for singular contact structures, and show we can
realise all desired boundary conditions except for normal anchoring on a
sphere, where a theorem of Eliashberg and Thurston provides an obstruction to
having a singular contact structure in the interior. By introducing a singular
version of the Lutz twist we show that all contact structures are homotopic
within the larger class of singular contact structures. We give applications of
our results to the description of topological defects in chiral liquid
crystals.
