Have very many linguists studied the structure of sentence fragments used in informal conversation in a given language?
In the course of informal conversation, we English speakers use lots of sentence fragments to answer questions (e.g. (It's) "In the barn," (It's) "The roses." (You can do/have/etc.) "Whatever you want." We also use fragments for other purposes, as in this fragment: (Look out,) "The wolf!" (is back). I imagine that the same is true in most languages.
However, some sentence fragments are clearly not permitted. For example, we English speakers can't say "Take the." (peas) for example.
Since linguists professional linguists can document the types of structures that are and aren't permitted in a language, I wonder which syntacticians have studied sentence fragments in informal conversation, how much work has been done in this area.