  This paper is devoted to the resolution of singularities of holomorphic
vector fields and of one-dimensional holomorphic foliations in dimension 3 and
it has two main objectives. First, from the general perspective of
one-dimensional foliations, we build upon the work of Cano-Roche-Spivakovsky
and essentially complete it. As a consequence, we obtain a general resolution
theorem comparable to the resolution theorem of McQuillan-Panazzolo but proved
by means of rather different methods.
  The second objective of this paper consists of looking at a special class of
singularities of foliations containing, in particular, all singularities of
complete holomorphic vector fields on complex manifolds of dimension 3. We then
prove that for this class of holomorphic foliations, there holds a much sharper
resolution theorem. This second result was the initial motivation of this paper
and it relies on the combination of the previous resolution theorems for
(general) foliations with some classical material on asymptotic expansions for
solutions of differential equations.
