  Painlev\'e showed that there can be inconsistency and indeterminacy in
solutions to the equations of motion of a 2D rigid body moving on a
sufficiently rough surface. The study of Painlev\'e paradoxes in 3D has
received far less attention. In this paper, we highlight the pivotal role in
the dynamics of the azimuthal angular velocity Psi by proving the existence of
three critical values of Psi, one of which occurs independently of any paradox.
We show that the 2D problem is highly singular and uncover a rich geometry in
the 3D problem which we use to explain recent numerical results.
