Consider generic configurations consisting of 4 distinct circles on the sphere.
Two configurations are equivalent if they can be mapped onto each other by a homeomorphism of the sphere. How to enumerate/classify such configurations?

Equivalent problem: classify the arrangements of 4 hyperbolic planes in the hyperbolic space, up to homeomorphisms of the space.

Before voting to close this question as trivial, you may look at the <a href="http://www.math.purdue.edu/~eremenko/dvi/4circles-generic.pdf">classification of generic configurations</a> which we obtained by brute force:

[![configurations without disjoint pairs of circles][1]][1]

[![configurations with at least one disjoint pair of circles][2]][2]

Each region bounded by more than 3 sides is labeled by the number of its boundary sides. This is used to show that all configurations are non-equivalent.

Questions: Is this new? Is there a scientific method to obtain this? Is there any structure on these 35 configurations?

There is a large research area about hyperplane arrangements in a Euclidean space. How about hyperbolic space? There is also a large body of research on
hyperbolic tetrahedra. But it is always assumed that the tetrahedron is compact (or has only vertices at infinity).

We encountered this question in <a href="http://www.math.purdue.edu/~eremenko/dvi/pent18.pdf">our studies of the Heun and Painlevé VI equations</a> with real coefficients. (See Appendix II in the linked paper).
Projective monodromy groups associated to these equations are generated by 4 reflections in circles.

EDIT. It seems that the problem is of purely topological nature: for any collection of Jordan curves on the sphere, such that each pair intersects transversally at at most two points, there exists an equivalent configuration of circles:
A. Bobenko, B. Springborn, [Variational principles for circle patterns and Koebe's theorem](https://doi.org/10.1090/S0002-9947-03-03239-2 ). Trans. Amer. Math. Soc. 356 (2004), no. 2, 659–689. 

EDIT2. The previous remark is incorrect (thanks to Ivan Izmestiev for his comment). A counterexample with 5 curves is contained in this paper:
MR3216670 Kang, Ross J.; Müller, Tobias [Arrangements of pseudocircles and circles](https://doi.org/10.1007/s00454-014-9583-8), Discrete Comput. Geom. 51 (2014), no. 4.

EDIT3. From the [very last Notices AMS](https://www.ams.org/journals/notices/201809/rnoti-p1062.pdf) (October 18) I learned that the number of
arrangements of $n$ circles on the plane has been studied, and that this number even has a name [A250001](https://oeis.org/A250001) in the Online encyclopedia of integer sequences.
However, even for n=4 it is not easy to derive the result for the sphere from the result for the plane (the only way to do this that I see is by direct examination of of equivalences of configurations).
 
  [1]: https://i.sstatic.net/Qul0y.png
  [2]: https://i.sstatic.net/cnhTI.png