  We discuss various novel features of $n(\ge 4)$-dimensional spacetimes
sourced by a massless (non-)phantom scalar field in general relativity.
Assuming that the metric is a warped product of static two-dimensional
Lorentzian spacetime and an $(n-2)$-dimensional Einstein space $K^{n-2}$ with
curvature $k=0, \pm 1$, and that the scalar field depends only on the radial
variable, we present a complete classification of static solutions for both
signs of kinetic term. Contrary to the case with a non-phantom scalar field,
the Fisher solution is not unique, and there exist two additional metrics
corresponding to the generalizations of the Ellis-Gibbons solution and the
Ellis-Bronnikov solution. We explore the maximal extension of these solutions
in detail by the analysis of null/spacelike geodesics and singularity. For the
phantom Fisher and Ellis-Gibbons solutions, we find that there inevitably
appear parallelly propagated (p.p) curvature singularities in the parameter
region where there are no scalar curvature singularities. Interestingly, the
areal radius blows up at these p.p curvature singularities, which are
nevertheless accessible within a finite affine time along the radial null
geodesics. It follows that only the Ellis-Bronnikov solution describes a
regular wormhole in the two-sided asymptotically flat spacetime. Using the
general transformation relating the Einstein and Jordan frames, we also present
a complete classification of solutions with the same symmetry coupled to a
conformal scalar field. Additionally, by solving the field equations in the
Jordan frame, we prove that this classification is genuinely complete.
