I enumerate all possible types of spacetime causal structures that can appear in static, spherically symmetric configurations of a self-gravitating, real, nonlinear, minimally coupled scalar field φ in general relativity, with an arbitrary potential V(φ), not necessarily positive definite. It is shown that a variable scalar field adds nothing to the list of possible structures with a constant φ field, namely, Minkowski [or anti-de Sitter (AdS)], Schwarzschild, de Sitter, and Schwarzschild-de Sitter structures. It follows in particular that whatever V(φ) is, this theory does not admit regular black holes with flat or AdS asymptotics. It is concluded that the only possible globally regular, asymptotically flat solutions are solitons with a regular center, without horizons and with at least partly negative potentials V(φ). Extension of the results to more general field models is discussed. ©2001 The American Physical Society.