  We show that codimension one distributions with at most isolated
singularities on certain smooth projective threefolds with Picard rank one have
stable tangent sheaves. The ideas in the proof of this fact are then applied to
the characterization of certain irreducible components of the moduli space of
stable rank 2 reflexive sheaves on $\mathbb{P}^3$, and to the construction of
stable rank 2 reflexive sheaves with prescribed Chern classes on general
threefolds. We also prove that if $\mathscr{G}$ is a subfoliation of a
codimension one distribution $\mathscr{F}$ with isolated singularities, then
$Sing(\mathscr{G})$ is a curve. As a consequence, we give a criterion to decide
whether $\mathscr{G}$ is globally given as the intersection of $\mathscr{F}$
with another codimension one distribution. Turning our attention to codimension
one distributions with non isolated singularities, we determine the number of
connected components of the pure 1-dimensional component of the singular
scheme.
