Yes, this has been done in other settings. For example, Scott Balchin, Kyle Ormsby, Angélica M. Osorno, and Constanze Roitzheim wrote a paper, [Model structures on finite total orders][1], that enumerates all model structures on a finite total order [n]. In fact, this paper is one in a series of recent papers on problems like this, usually with at least one of those four authors involved. For instance, the paper [Composition closed premodel structures and the Kreweras lattice][2] counts premodel structures and identifies model structures with certain tricolored trees. This area is starting to be called [homotopical combinatorics][3], and at that link you can read a nice description by Kyle Ormsby about the connection between lattices and weak factorization systems.

I believe Inna Zakharevich had done earlier work counting model structures on posets, but I need to take some time to search for it. And, I think Andrew Salch told me about some work of his related to the nine model structures problem, perhaps extending it to vector spaces. I will look into my notes when I have time, and might edit this answer with more references if I find them.


  [1]: https://arxiv.org/abs/2109.07803
  [2]: https://arxiv.org/abs/2209.03454
  [3]: https://kyleormsby.github.io/posts/2021/09/homotopical-combinatorics/