  We study the conormal sheaves and singular schemes of 1-dimensional
foliations on smooth projective varieties $X$ of dimension 3 and Picard rank 1.
We prove that if the singular scheme has dimension 0, then the conormal sheaf
is $\mu$-stable whenever the tangent bundle $TX$ is stable, and apply this fact
to the characterization of certain irreducible components of the moduli space
of rank 2 reflexive sheaves on $\mathbb{P}^3$ and on a smooth quadric
hypersurface $Q_3\subset\mathbb{P}^4$. Finally, we give a classification of
local complete intersection foliations, that is, foliations with locally free
conormal sheaves, of degree 0 and 1 on $Q_3$.
