  We study the tangential Poisson cohomology (TP-cohomology) of regular Poisson
manifolds, first defined by Lichnerowicz using contravariant tensor fields. We
show that for a regular Poisson manifold M, the TP-cohomology coincides with
the leafwise de Rham (or Cech) cohomology of the symplectic foliation of M. Its
computation in various degrees leads to open, non trivial problems. To get a
better understanding of these difficulties, we study explicitly many examples
coming from nilpotent and 3-dimensional (real) Lie algebras. For the latter, we
compare the TP-cohomology and the usual Poisson cohomology (P-cohomology).
