We construct families of inextendible spacetimes (M, g, Φ) where (g, Φ) satisfy the Einstein conformal scalar equations and the factor 1 − (8πk/6)Φ² vanishes on a regular hypersurface S of M. Despite the fact that the dynamical equations degenerate on such solutions, nevertheless, the curvature for all constructed (M, g, Φ) is regular in an open vicinity containing S. The spacetimes constructed and discussed in this work fall into two classes: the first class includes spacetimes where in a maximal analytical extension of the originally geodesically incomplete (M, g, Φ), the field Φ diverges on a regular null hypersurface, while the second class involves spacetimes where the divergence of Φ takes place on a regular null hypersurface constituting the boundary of the spacetime. The first family includes the well-known Bocharova–Bronikov–Melnikov–Bekenstein family but in addition includes a family of cosmological spacetimes while the second class describes local (non-vacuum) Einstein–Rosen waves. Consequences of the divergent behaviour of the field Φ on a smooth null hypersurface and its bearing on the physical interpretation of those spacetimes are also briefly addressed.