By a rational homology 3-sphere, I mean a compact oriented manifold three-manifold $Y$ with $H\_1(Y)$ finite. My question is whether there exists a reasonable classification of such manifolds such that $\pi\_1(Y)$ has a minimal presentation with two generators? How about three generators? 


Here are the examples I know:


Example 1. Connected sum of lens spaces
Example 2. Poincare sphere (Thanks to Qiaochu Yuan for pointing out in a previous version of this question that the Poincare sphere satisfies these criteria).



Question about the fundamental group of rational homology 3-spheres
As indicated in the comments, the greatest class of rank 2 3-manifolds (including rational homology spheres) are the genus 2 manifolds, which as indicated by Ruberman are double branched covers over links (coming from the hyperelliptic involution of the genus 2 surface which extends over both handlebodies). This includes connect sums of lens spaces and all the spherical space forms, as you indicated. However, it was a celebrated result of [Boileau and Zieschang](http://link.springer.com/article/10.1007%2FBF01388469) that there are certain Seifert-fibered spaces with rank 2 fundamental group, but Heegaard genus $=3$. They completely classified the rank 2 Seifert fibered spaces. These examples were extended by [Weidmann](http://link.springer.com/article/10.1007%2FBF01388469) and [Schultens-Weidmann](http://msp.org/pjm/2007/231-2/p12.xhtml) to certain graph manifolds (with rank $2$ fundamental group and genus $>2$). I think the state of the art in the non-hyperbolic case is a partial classification theorem of [Boileau-Weidmann](http://www.sciencedirect.com/science/article/pii/S0040938304000825). 


It is still unknown whether 2-generated hyperbolic 3-manifolds have Heegaard genus 2. Thurston observed that such manifolds are also branched covers over a link in $S^3$. Also, certain finiteness results are known for hyperbolic manifolds with rank 2 fundamental group, genus $>2$, and injectivity radius bounded below which has been [written by Souto](http://www.msp.warwick.ac.uk/gtm/2007/12/p015.xhtml). 


