I will explain what is a contact Fano manifold X and why we are
interested in studying it. The main conjecture in the subject (due to
LeBrun and Salamon) is that X is always a homogeneous space, one of the
adjoint varieties. The main tool from algebraic geometry to study the
conjecture is the theory of minimal rational curves, i.e., in this case,
the rational curves  on X with minimal degree measured by the
intersection with the anti-canonical divisor (they are called lines, or
contact lines). A general line is smooth and standard, i.e,. it has a
particularly simple splitting of the tangent bundle to X restricted to
the curve. Working on contact Fano manifolds would go very smoothly, if
all lines were smooth and standard. Although we cannot prove that, we
managed to bound the dimensions of spaces of singular and non-standard
lines. Specifically:
1) The dimension of the space parametrising singular lines is at most
around 2/3 of the dimension of the space parametrising all lines.
2) The codimension of the space parametrising non-standard lines is at
least 2.
The second statement was earlier claimed by Kebekus and used by him to
prove an irreducibility of the space of lines through a fixed general
point. However, his proof of 2) contained a gap.