  With the intent of laying the groundwork for a program that aims at
explicitly describing the space of Cartan (i.e. multiplicative) connections on
a general proper Lie groupoid, we begin to investigate the space of such
connections in the regular case. We point out that there is a close
relationship between Cartan connections on a proper regular groupoid and
representations of the groupoid on its own longitudinal bundle (i.e. on the
vector distribution tangent to its orbits). This observation enables us to
reduce the original problem to a simpler one. We carry out a prospective study
of the latter problem, and apply the resulting analysis to produce a number of
examples in rank two which serve to illustrate the diversity of the possible
obstructions to the existence of multiplicative connections.
