  For a possibly singular subset of a regular Poisson manifold we construct a
deformation quantization of its algebra of Whitney functions. We then extend
the construction of a deformation quantization to the case where the underlying
set is a subset of a not necessarily regular Poisson manifold which can be
written as the quotient of a regular Poisson manifold on which a compact Lie
group acts freely by Poisson maps. Finally, if the quotient Poisson manifold is
regular as well, we show a "quantization commutes with reduction" type result.
For the proofs, we use methods stemming from both singularity theory and
Poisson geometry.
