  Poisson algebras are, just like Lie algebras, particular cases of
Lie-Rinehart algebras. The latter were introduced by Rinehart in his seminal
1963 paper, where he also introduces the notion of an enveloping algebra and
proves --- under some mild conditions --- that the enveloping algebra of a
Lie-Rinehart algebra satisfies a Poincar\'e-Birkhoff-Witt theorem (PBW
theorem). In the case of a Poisson algebra $({\mathcal
A},\cdot,\{\cdot,\cdot\})$ over a commutative ring $R$ (with unit), Rinehart's
result boils down to the statement that if $\mathcal A$ is \emph{smooth} (as an
algebra), then gr$(U({\mathcal A}))$ and $\mathrm{Sym}_{\mathcal
A}(\Omega({\mathcal A}))$ are isomorphic as graded algebras; in this formula,
$U({\mathcal A})$ stands for the Poisson enveloping algebra of ${\mathcal A}$
and $\Omega({\mathcal A})$ is the ${\mathcal A}$-module of K\"ahler
differentials of ${\mathcal A}$ (viewing ${\mathcal A}$ as an $R$-algebra). In
this paper, we give several new constructions of the Poisson enveloping algebra
in some general and in some particular contexts. Moreover, we show that for an
important class of \emph{singular} Poisson algebras, the PBW theorem still
holds. In geometrical terms, these Poisson algebras correspond to (singular)
Poisson hypersurfaces of arbitrary smooth affine Poisson varieties.
