Each foliation of 3-dimensional Euclidean space by pairwise nonparallel lines extends to a foliation of real projective 3-space by nonintersecting projective lines. Hence lifts to a foliation of the 3-sphere by nonintersecting great circles. So the complete classification of all foliations of 3-dimensional Euclidean space by pairwise nonparallel lines is the classification of great circle fibrations of the 3-sphere, which is worked out very clearly in Gluck, Warner, Great circle fibrations of the 3-sphere, Duke Math. J., Volume 50, Number 1 (1983), 107-132.

Each foliation of 3-dimensional Euclidean space by pairwise nonparallel lines extends to a foliation of real projective 3-space by nonintersecting projective lines. Hence lifts to a foliation of the 3-sphere by nonintersecting great circles. So the complete classification of all foliations of 3-dimensional Euclidean space by pairwise nonparallel lines is the classification of great circle fibrations of the 3-sphere, which is worked out very clearly in Gluck, Warner, Great circle fibrations of the 3-sphere, Duke Math. J., Volume 50, Number 1 (1983), 107-132.

To give some idea of the method: each great circle spans an unique oriented 2-plane through the origin of $\mathbb{R}^4$, which is spanned by an oriented orthonormal basis $u,v$. The 2-vector $\zeta=u\wedge v$ is uniquely determined. It splits into a self-dual and an anti-self-dual part, say $\zeta_+,\zeta_-$, each of the same length, which we rescale to be unit length.

For a great circle fibration, every choice of $\zeta_-$ occurs uniquely, i.e. for a unique circle fiber, so $\zeta_+$ can be written as a function of $\zeta_-$. This functions turns out to be smooth, a smooth map from the unit sphere in the anti-self-dual 2-vectors to the unit sphere in the self-dual. This map is strictly contracting in the usual metric on those spheres. Moreover, any strictly contracting map arises in this way. All of this is proven in great detail in the paper.