  We construct asymptotically flat, spinning, regular on and outside an event
horizon, scalarised black holes (SBHs) in extended scalar-tensor-Gauss-Bonnet
models. They reduce to Kerr BHs when the scalar field vanishes. For an
illustrative choice of non-minimal coupling, we scan the domain of existence.
For each value of spin, SBHs exist in an interval between two critical masses,
with the lowest one vanishing in the static limit. Non-uniqueness with Kerr BHs
of equal global charges is observed; the SBHs are entropically favoured. This
suggests SBHs form dynamically from the spontaneous scalarisation of Kerr BHs,
which are prone to a scalar-triggered tachyonic instability, below the largest
critical mass. Phenomenologically, the introduction of BH spin damps the
maximal observable difference between comparable scalarised and vacuum BHs. In
the static limit, (perturbatively stable) SBHs can store over 20% of the
spacetime energy outside the event horizon; in comparison with Schwarzschild
BHs, their geodesic frequency at the ISCO can differ by a factor of 2.5 and
deviations in the shadow areal radius may top 40%. As the BH spin grows, low
mass SBHs are excluded, and the maximal relative differences decrease, becoming
of order $\sim$ few % for dimensionless spin $j\gtrsim 0.5$. This reveals a
spin selection effect: non-GR effects are only significant for low spin. We
discuss if and how the recently measured shadow size of the M87 supermassive
BH, constrains the length scale of the Gauss-Bonnet coupling.
