  Extended scalar-tensor-Gauss-Bonnet (eSTGB) gravity has been recently argued
to exhibit spontaneous scalarisation of vacuum black holes (BHs). A similar
phenomenon can be expected in a larger class of models, which includes e.g.
Einstein-Maxwell-scalar (EMS) models, where spontaneous scalarisation of
electrovacuum BHs should occur. EMS models have no higher curvature
corrections, a technical simplification over eSTGB models that allows us to
investigate, fully non-linearly, BH scalarisation in two novel directions.
Firstly, numerical simulations in spherical symmetry show, dynamically, that
Reissner-Nordstr\"om (RN) BHs evolve into a perturbatively stable scalarised
BH. Secondly, we compute the non-spherical sector of static scalarised BH
solutions bifurcating from the RN BH trunk. Scalarised BHs form an infinite
(countable) number of branches, and possess a large freedom in their multipole
structure. Unlike the case of electrovacuum, the EMS model admits static,
asymptotically flat, regular on and outside the horizon BHs without spherical
symmetry and even without any spatial isometries, which are thermodynamically
preferred over the electrovacuum state. We speculate on a possible dynamical
role of these non-spherical scalarised BHs.
