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A Q-homology plane is a normal complex algebraic surface having trivialrational homology. We classify singular Q-homology planes which are C^1- orC*-ruled. We analyze their completions, the number of different rulings, thenumber of affine lines on it and we give constructions. Together withpreviously known results this completes the classification of Q-homology planeswith smooth locus of non-general type. We show also that the dimension of afamily of homeomorphic but non-isomorphic singular Q-homology planes having thesame weighted boundary, singularities and Kodaira dimension can be arbitrarilybig.

Classification of singular Q-homology planes. II. C^1- and C*-rulings