Classification of singular Q-homology planes. II. C¹- and C*-rulings

A Q-homology plane is a normal complex algebraic surface having trivial rational homology. We classify singular Q-homology planes which are C^1- or C*-ruled. We analyze their completions, the number of different rulings, the number of affine lines on it and we give constructions. Together with previously known results this completes the classification of Q-homology planes with smooth locus of non-general type. We show also that the dimension of a family of homeomorphic but non-isomorphic singular Q-homology planes having the same weighted boundary, singularities and Kodaira dimension can be arbitrarily big.