  Electro-vacuum black holes are scale-invariant; their energy-momentum tensor
is traceless. Quantum corrections of various sorts, however, can often produce
a trace anomaly and a breakdown of scale-invariance. The (quantum-corrected)
black hole solutions of the corresponding gravitational effective field theory
(EFT) have a non-vanishing Ricci scalar. Then, the presence of a scalar field
with the standard non-minimal coupling $\xi \phi^2 R$ naturally triggers a
spontaneous scalarisation of the corresponding black holes. This scalarisation
phenomenon occurs for an (infinite) discrete set of $\xi$. We illustrate the
occurrence of this phenomenon for two examples of static, spherically
symmetric, asymptotically flat black hole solution of EFTs. In one example the
trace anomaly comes from the matter sector -- a novel, closed form,
generalisation of the Reissner-Nordstr\"om solution with an $F^4$ correction --
whereas in the other example it comes from the geometry sector -- a
noncommutative geometry generalization of the Schwarzschild black hole. For
comparison, we also consider the scalarisation of a black hole surrounded by
(non-conformally invariant) classical matter (Einstein-Maxwell-dilaton black
holes). We find that the scalarised solutions are, generically, entropically
favoured.
