  We test the stability of various wormholes and black holes supported by a
scalar field with a negative kinetic term. The general axial perturbations and
the monopole type of polar perturbations are considered in the linear
approximation. Two classes of objects are considered: (i) wormholes with flat
asymptotic behavior at one end and AdS on the other (M-AdS wormholes) and (ii)
regular black holes with asymptotically de Sitter expansion far beyond the
horizon (the so-called black universes). A difficulty in such stability studies
is that the effective potential for perturbations forms an infinite wall at
throats, if any. Its regularization is in general possible only by numerical
methods, and such a method is suggested in a general form and used in the
present paper. As a result, we have shown that all configurations under study
are unstable under spherically symmetric perturbations, except for a special
class of black universes where the event horizon coincides with the minimum of
the area function. For this stable family, the frequencies of quasinormal modes
of axial perturbations are calculated.
