  In the projective space $\mathrm{PG}(3,q)$, we consider the orbits of lines
under the stabilizer group of the twisted cubic. In the literature, lines of
$\mathrm{PG}(3,q)$ are partitioned into classes, each of which is a union of
line orbits. In this paper, all classes of lines consisting of a unique orbit
are found. For the remaining line types, with one exception, it is proved that
they consist exactly of two or three orbits; sizes and structures of these
orbits are determined. Also, the subgroups of the stabilizer group of the
twisted cubic fixing lines of the orbits are obtained. Problems which remain
open for one type of lines are formulated and, for $5\le q\le37$ and $q=64$, a
solution is provided.

Combinatorics