  Black hole `spectroscopy', i.e. the identification of quasinormal mode
frequencies via gravitational wave observations, is a powerful technique for
testing the general relativistic nature of black holes. In theories of gravity
beyond general relativity perturbed black holes are typically described by a
set of coupled wave equations for the tensorial field and the extra
scalar/vector degrees of freedom, thus leading to a theory-specific quasinormal
mode spectrum. In this paper we use the eikonal/geometric optics approximation
to obtain analytic formulae for the frequency and damping rate of the
fundamental quasinormal mode of a generalised, theory-agnostic system of
equations describing coupled scalar-tensor perturbations of spherically
symmetric black holes. Representing an extension of our recent work, the
present model includes a massive scalar field, couplings through the field
derivatives and first-order frame dragging rotational corrections. Moving away
from spherical symmetry, we consider the simple model of the scalar wave
equation in a general stationary-axisymmetric spacetime and use the eikonal
approximation to compute the quasinormal modes associated with equatorial and
nonequatorial photon rings.
