  We study the stability of static, spherically symmetric solutions to the
Einstein equations with a scalar field as the source. We describe a general
methodology of studying small radial perturbations of scalar-vacuum
configurations with arbitrary potentials V(\phi), and in particular space-times
with throats (including wormholes), which are possible if the scalar is
phantom. At such a throat, the effective potential for perturbations V_eff has
a positive pole (a potential wall) that prevents a complete perturbation
analysis. We show that, generically, (i) V_eff has precisely the form required
for regularization by the known S-deformation method, and (ii) a solution with
the regularized potential leads to regular scalar field and metric
perturbations of the initial configuration. The well-known conformal mappings
make these results also applicable to scalar-tensor and f(R) theories of
gravity. As a particular example, we prove the instability of all static
solutions with both normal and phantom scalars and V(\phi) = 0 under spherical
perturbations. We thus confirm the previous results on the unstable nature of
anti-Fisher wormholes and Fisher's singular solution and prove the instability
of other branches of these solutions including the anti-Fisher "cold black
holes".
