  We show that, independently of the scalar field potential and of specific
asymptotic properties of the spacetime (asymptotically flat, de Sitter or
anti-de Sitter), any static, spherically symmetric or planar, black hole or
soliton solution of the Einstein theory minimally coupled to a real scalar
field with a general potential is mode stable under linear odd-parity
perturbations. To this end, we generalize the Regge-Wheeler equation for a
generic self-interacting scalar field, and show that the potential of the
relevant Schr\"odinger operator can be mapped, by the so-called S-deformation,
to a semi-positively defined potential. With these results at hand we study the
existence of slowly rotating configurations. The frame dragging effect is
compared with the Kerr black hole.
