We study self-gravitating, static, spherically symmetric phantom scalar fields with arbitrary potentials (favored by cosmological observations) and single out 16 classes of possible regular configurations with flat, de Sitter, and anti-de Sitter asymptotics. Among them are traversable wormholes, bouncing Kantowski-Sachs (KS) cosmologies, and asymptotically flat black holes (BHs). A regular BH has a Schwarzschild-like causal structure, but the singularity is replaced by a de Sitter infinity, giving a hypothetic BH explorer a chance to survive. It also looks possible that our Universe has originated in a phantom-dominated collapse in another universe, with KS expansion and isotropization after crossing the horizon. Explicit examples of regular solutions are built and discussed. Possible generalizations include k-essence type scalar fields (with a potential) and scalar-tensor gravity.