  A Q-homology plane is a normal complex algebraic surface having trivial
rational homology. We obtain a structure theorem for Q-homology planes with
smooth locus of non-general type. We show that if a Q-homology plane contains a
non-quotient singularity then it is a quotient of an affine cone over a
projective curve by an action of a finite group respecting the set of lines
through the vertex. In particular, it is contractible, has negative Kodaira
dimension and only one singular point. We describe minimal normal completions
of such planes.
