  A fibration of $\mathbb{R}^3$ by oriented lines is given by a unit vector
field $V : \mathbb{R}^3 \to S^2$, for which all of the integral curves are
oriented lines. A line fibration is called skew if no two fibers are parallel.
Skew fibrations have been the focus of recent study, in part due to their close
relationships with great circle fibrations of $S^3$ and with tight contact
structures on $\mathbb{R}^3$. Both geometric and topological classifications of
the space of skew fibrations have appeared; these classifications rely on
certain rigid geometric properties exhibited by skew fibrations. Here we study
these properties for line fibrations which are not necessarily skew, and we
offer some partial answers to the question: in what sense do nonskew fibrations
look and behave like skew fibrations? We develop and utilize a technique,
called the parallel plane pushoff, for studying nonskew fibrations. In
addition, we summarize the known relationship between line fibrations and
contact structures, and we extend these results to give a complete
correspondence. Finally, we develop a technique for generating nonskew
fibrations and offer a number of examples.
