  In a series of two articles Kebekus studied deformation theory of minimal
rational curves on contact Fano manifolds. Such curves are called contact
lines. Kebekus proved that a contact line through a general point is
necessarily smooth and has a fixed standard splitting type of the restricted
tangent bundle. In this paper we study singular contact lines and those with
special splitting type. We provide restrictions on the families of such lines,
and on contact Fano manifolds which have reducible varieties of minimal
rational tangents. We also show that the results about singular lines naturally
generalise to complex contact manifolds, which are not necessarily Fano, for
instance, quasi-projective contact manifolds or compact contact manifolds of
Fujiki class C. In particular, in many cases the dimension of a family of
singular lines is at most 2 less than the dimension of the contact manifold.
