  The imminent detection of gravitational waves will trigger precision tests of
gravity through observations of quasinormal ringing of black holes. While
General Relativity predicts just two polarizations of gravitational waves, the
so-called plus and cross polarizations, numerous alternative theories of
gravity predict up to six different polarizations which will potentially be
observed in current and future generations of gravitational wave detectors.
Bekenstein's Tensor-Vector-Scalar (TeVeS) theory and its generalization fall
into one such class of theory that predict the full gamut of six polarizations
of gravitational waves. In this paper we begin the study of quasinormal modes
(QNMs) in TeVeS by studying perturbations of the scalar field in a spherically
symmetric background. We show that, at least in the case where superluminal
propagation of perturbations is not present, black holes are generically stable
to this kind of perturbation. We also make a unique prediction that, as the
limit of the various coupling parameters of the theory tend to zero, the QNM
spectrum tends to $1/\sqrt{2}$ times the QNM spectrum induced by scalar
perturbations of a Schwarzschild black hole in General Relativity due to the
intrinsic presence of the background vector field. We further show that the QNM
spectrum does not vary significantly from this value for small values of the
theory's coupling parameters, however can vary by as much as a few percent for
larger, but still physically relevant parameters.
