  We perform a fully relativistic analysis of odd-type linear perturbations
around a static and spherically symmetric solution in the most general
scalar-tensor theory with second-order field equations in four-dimensional
spacetime. It is shown that, as in the case of general relativity, the
quadratic action for the perturbations reduces to the one having only a single
dynamical variable, from which concise formulas for no-ghost and no-gradient
instability conditions are derived. Our result is applicable to all the
theories of gravity with an extra scalar degree of freedom. We demonstrate how
the generic formulas can be applied to some particular examples such as the
Brans-Dicke theory, $f(R)$ models, and Galileon gravity.
