  We characterize the square-integrable representations of (connected, simply
connected) solvable Lie groups in terms of the generalized orbits of the
coadjoint action. We prove that the normal representations corresponding, via
the Puk\'anszky correspondence, to open coadjoint orbits are type I, not
necessarily square-integrable representations. We show that the
quasi-equivalence classes of type I square-integrable representations are in
bijection with the simply connected open coadjoint orbits, and the existence of
an open coadjoint orbit guarantees the existence of a compact open subset of
the space of primitive ideals of the group. When the nilradical has codimension
1, we prove that the isolated points of the primitive ideal space are always of
type I. This is not always true for codimension greater than 2, as shown by
specific examples of solvable Lie groups that have dense, but not locally
closed, coadjoint orbits.
