The Gowdy spacetimes are vacuum solutions of the Einstein equations with two commuting Killing vectors having compact spacelike orbits with T³, S² × S¹ or S³ topology. In the case of T³ topology, Kichenassamy and Rendall have found a family of singular solutions which are asymptotically velocity dominated by construction. In the case when the velocity is between 0 and 1, the solutions depend on the maximal number of free functions. We consider the similar case with S² × S¹ or S³ topology, where the main complication is the presence of symmetry axes. The results for T³ may be applied locally except at the axes, where one of the Killing vectors degenerates. We use Fuchsian techniques to show the existence of singular solutions similar to the T³ case. We first solve the analytic case and then generalize to the smooth case by approximating smooth data with a sequence of analytic data. However, for the metric to be smooth at the axes, the velocity must be −1 or 3 there, which is outside the range where the constructed solutions depend on the full number of free functions. A plausible explanation is that in general a spiky feature may develop at the axis, a situation which is unsuitable for a direct treatment by Fuchsian methods.