  Recent studies have revealed an oscillating asymmetry in the vertical
structure of the Milky Way's disc. Here we analyze 16 high-resolution, fully
cosmological simulations of the evolution of individual Milky Way-sized
galaxies, carried out with the MHD code AREPO. At redshift zero, about $70\%$
of our galactic discs show strong vertical patterns, with amplitudes that can
exceed 2 kpc. Half of these are typical `integral sign' warps. The rest are
oscillations similar to those observed in the Milky Way. Such structures are
thus expected to be common. The associated mean vertical motions can be as
large as 30~km/s. Cold disc gas typically follows the vertical patterns seen in
the stars. These perturbations have a variety of causes: close encounters with
satellites, distant flybys of massive objects, accretion of misaligned cold gas
from halo infall or from mergers. Tidally induced vertical patterns can be
identified in both young and old stellar populations, whereas those originating
from cold gas accretion are seen mainly in the younger populations. Galaxies
with regular or at most weakly perturbed discs are usually, but not always,
free from recent interactions with massive companions, although we have one
case where an equilibrium compact disc reforms after a merger.
