We consider two incidence problems for integral curves of vector fields. The first is an analogue of the Euclidean joints problem, in which lines are replaced by integral curves of smooth vector fields taken from some finite-dimensional set. The second is a bilinear and more rigid version of the first, in which we have two vector fields and one family of integral curves tangent to each and wish to know how many intersecting pairs are possible. In both cases, a curvature condition of Hörmander makes possible nontrivial bounds on the number of incidences in terms of the number of curves.