One common scenario is that I'm working on one problem, and in the process of solving that issue do some refactoring of related code. In this case, I want to commit the refactoring (which does not change the program's behaviour) before committing the changes that do change the program's behaviour.
I typically then send that first refactoring commit to Github (on its own branch) so that it gets full CI test coverage. And then continue working on the fix/feature while it runs.
