  We study foliations by curves on the three-dimensional projective space with
no isolated singularities, which is equivalent to assuming that the conormal
sheaf is locally free. We provide a classification of the topological and
algebraic invariants of the conormal sheaves and singular schemes for such
foliations by curves, up to degree 3. In particular, we prove that foliations
by curves of degree 1 or 2 are contained in a pencil of planes or are
Legendrian, and are given by the complete intersection of two codimension one
distributions. Furthermore, we prove that the conormal sheaf of a foliation by
curves of degree 3 with reduced singular scheme either splits as a sum of line
bundles or is an instanton bundle. For degree larger than 3, we focus on two
classes of foliations by curves, namely Legendrian foliations and those whose
conormal sheaf is a twisted null-correlation bundle. We give characterizations
of such foliations, describe their singular schemes and their moduli spaces.
