The free discrete subgroups consist of the closure of the Riley slice of Schottky space (notice that one may assume $Re(\alpha)\geq 0, Im(\alpha)\geq 0$, since $\alpha \mapsto -\alpha, \overline{\alpha}$ preserves discreteness). 
Here's a picture of the first quadrant of the Riley slice:
![enter image description here][1]

The exterior of the black fractal represents free discrete groups that are generalized Schottky groups. These have been investigated in detail by [Akiyoshi, Sakuma, Wada, and Yamashita][2].   The black curve consists of geometrically finite groups with a cusp or degenerate groups. By the [density conjecture][3], it is known that all free two-parabolic generator groups lie in the boundary of the Riley slice. Moreover, [it is known that there is a group realizing each ending lamination][4] parameterized by $\mathbb{R/Z}-\mathbb{Q/Z}$. 

In the exterior of the Riley slice, there are many more non-free discrete groups, such as those corresponding to 2-bridge links. In fact, [all of the torsion-free discrete groups correspond to 2-bridge links][5]. Note that for 2-bridge links, although they are known to be generated by 2 parabolics, the precise value of $\alpha$ which gives the parabolics has not been determined. One may see an example of this for the twist knots, for which $\alpha$ has been worked out by [Hoste and Shanahan][6]. 


For the discrete groups with torsion, see also some [incomplete notes of mine][7].


  [1]: https://i.sstatic.net/IqO9w.jpg
  [2]: http://www.ams.org/mathscinet-getitem?mr=1698107
  [3]: http://link.springer.com/article/10.1007/s11511-012-0088-0
  [4]: http://www.ams.org/books/conm/510/10029
  [5]: http://www.ams.org/mathscinet-getitem?mr=1393562
  [6]: http://www.worldscientific.com/doi/abs/10.1142/S0218216501001049
  [7]: https://www.dropbox.com/sh/cxdm5tu5hsnho0i/oByjC-AcYN/parabolic