  This short note provides a quick introduction to relative canonical
resolutions of curves on rational normal scrolls. We present our
Macaulay2-package which computes the relative canonical resolution associated
to a curve and a pencil of divisors. Most of our experimental data can be found
on a dedicated webpage. We end with a list of conjectural shapes of relative
canonical resolutions. In particular, for curves of genus $g=n\cdot k +1$ and
pencils of degree $k$ for $n\ge 1$, we conjecture that the syzygy divisors on
the Hurwitz space $\mathscr{H}_{g,k}$ constructed by Deopurkar and Patel all
have the same support.
