  We model neutron stars as magnetised hybrid stars with an abrupt hadron-quark
phase transition in their cores, taking into account current constraints from
nuclear experiments and multi-messenger observations. We include magnetic field
effects considering the Landau level quantisation of charged particles and the
anomalous magnetic moment of neutral particles. We construct the magnetised
hybrid equation of state, and we compute the particle population, the matter
magnetisation and the transverse and parallel pressure components. We integrate
the stable stellar models, considering the dynamical stability for \emph{rapid}
or \emph{slow} hadron-quark phase conversion. Finally, we calculate the
frequencies and damping times of the fundamental and $g$ non-radial oscillation
modes. The latter, a key mode to learn about phase transitions in compact
objects, is only obtained for stars with slow conversions. For low magnetic
fields, we find that one of the objects of the GW170817 binary system might be
a hybrid star belonging to the slow extended stability branch. For magnetars,
we find that a stronger magnetic field always softens the hadronic equation of
state. Besides, only for some parameter combinations a stronger magnetic field
implies a higher hybrid star maximum mass. Contrary to previous results, the
incorporation of anomalous magnetic moment does not affect the studied
astrophysical quantities. We discuss possible imprints of the microphysics of
the equation of state that could be tested observationally in the future, and
that might help infer the nature of dense matter and hybrid stars.
