  We introduce the concept of $\varepsilon\,$-contact metric structures on
oriented (pseudo-)Riemannian three-manifolds, which encompasses the usual
Riemannian contact metric, Lorentzian contact metric and para-contact metric
structures, but which also allows the possibility for the Reeb vector field to
be null. We investigate in more detail this latter case, which we call null
contact structure. We observe that it is possible to extend in a natural and
meaningful way both the Sasaki and K-contact conditions for null-contact
structures, but we find that they are not equivalent conditions, in
contradistinction to the situation for non-lightlike Reeb vector fields.
Finally, we define the notion of $\varepsilon\eta\,$-Einstein structures and we
discover that appropriate direct products of these structures produce solutions
of six-dimensional minimal supergravity coupled to a tensor multiplet with
constant dilaton.
