[Famously](https://mathoverflow.net/a/29653/2362), there are exactly nine model structures on the category of sets, which are detailed [here](https://www.matem.unam.mx/~omar/notes/modelcatsets.html). In this case, one can exhaustively determine all six [weak factorization systems][1] and then see which ones fit together into model structures.

**Question 1:**
Are there other categories for which this program can be carried out? How many model structures does one have in these cases?

For instance,

 - pointed sets?
 - Some simple presheaf categories like the category of maps of sets? Or $G$-sets for a group $G$?

 - Vector spaces over a division ring $k$?
 - Chain complexes over $k$ (of fixed length? Bounded above / below? Unbounded?)
 - variants on the two previous where $k$ is a bit more complicated? Maybe abelian groups?
 - …

**Question 2:**
Is there an example of a complete and cocomplete category for which it’s possible to enumerate all model structures, but not feasible to enumerate all weak factorization systems?


  [1]: https://ncatlab.org/nlab/show/weak+factorization+system