  Einstein-Maxwell-dilaton theory is an interesting and well-motivated
theoretical laboratory to explore the impact of new fundamental degrees of
freedom in the context of testing the no-hair conjecture, due to the existence
of hairy black hole solutions together with the propagation of scalar, vector
and tensor modes. In this paper we compute the quasinormal mode spectrum of
static and slowly rotating black holes for generic values of the dilaton
coupling, within a weak electric charge approximation. Our results suggest that
these spacetimes are stable for generic values of the dilaton coupling and the
black hole charge. We also show that while gravitational modes are only weakly
affected by the coupling with the dilaton, the spectrum of electromagnetic
modes exhibits a more pronounced dilaton-dependent breaking of isospectrality
between the axial and polar sectors. We further show that the gravitational
quasinormal modes are well approximated by the properties of unstable null
circular geodesics in those spacetimes, while the treatment of electromagnetic
and scalar modes can be simplified by a suitably modified Dudley-Finley scheme
for the perturbed equations.
