  The question which motivates the article is the following: given a group
acting on a CAT(0) cube complex, how can we prove that it is acylindrically
hyperbolic? Keeping this goal in mind, we show a weak acylindricity of the
action on the contact graph associated to a CAT(0) cube complex, and we prove a
characterisation of WPD contracting isometries of CAT(0) cube complexes. As a
first consequence, we find alternative arguments to show several criteria which
were proved by Indira Chatterji and Alexandre Martin. Next, we show that if a
group acts essentially and acylindrically on the hyperplanes (i.e., the
intersection of the stabilisers of two hyperplanes which are sufficiently far
away from each other has its cardinality uniformly bounded) on a
finite-dimensional CAT(0) cube complex, then it contains a finite-index
subgroup which is either acylindrically hyperbolic or cyclic. Finally, we prove
a statement about codimension-one subgroups, which implies that, if a group
contains a finitely generated codimension-one subgroup which is malnormal and
which satisfies the bounded packing property, then this group must contain a
finite-index subgroup which is either acylindrically hyperbolic or cyclic.
