  We study the behavior of static, spherically symmetric solutions to the field
equations of the scalar-tensor theories (STT) of gravity belonging to the
Bergmann-Wagoner-Nordtvedt class, in the presence of an electric and/or
magnetic charge. The study is restricted to canonical (nonphantom) versions of
the theories and scalar fields without a self-interaction potential. Only
radial (monopole) perturbations are considered as the most likely ones to cause
an instability. The static background solutions contain naked singularities,
but we formulate the boundary conditions in such a way that would preserve
their meaning if a singularity is smoothed, for example, due to quantum gravity
effects. Since the solutions of all STT under study are related by conformal
transformations, the stability problem for all of them reduces to the same wave
equation, but the boundary conditions for perturbations (and sometimes the
boundaries themselves) are different in different STT, which affects the
stability results. The stability or instability conclusions are obtained for
different branches of the solutions in the Brans-Dicke, Barker and Schwinger
STT as well as for nonminimally coupled scalar fields with an arbitrary
parameter $\xi$ in the GR framework.
