  We instigate the angle-dependent magnetoresistance (AMR) of the layered
nodal-line Dirac semimetal ZrSiS for the in-plane and out-of-plane current
directions. This material has recently revealed an intriguing butterfly-shaped
in-plane AMR that is not well understood. Our measurements of the polar
out-of-plane AMR show a surprisingly different response with a pronounced
cusp-like feature. The maximum of the cusp-like anisotropy is reached when the
magnetic field is oriented in the $a$-$b$ plane. Moreover, the AMR for the
azimuthal out-of-plane current direction exhibits a very strong four-fold
$a$-$b$ plane anisotropy. Combining the Fermi surfaces calculated from first
principles with the Boltzmann's semiclassical transport theory we reproduce and
explain all the prominent features of the unusual behavior of the in-plane and
out-of-plane AMR. We are also able to clarify the origin of the strong
non-saturating transverse magnetoresistance as an effect of imperfect
charge-carrier compensation and open orbits. Finally, by combining our
theoretical model and experimental data we estimate the average relaxation time
of $2.6\times10^{-14}$~s and the mean free path of $15$~nm at 1.8~K in our
samples of ZrSiS.
