This is a follow-up of my previous question: Why do we need nil? Clearly, proper lists are used most of the time. But what is the purpose of an improper list?
For no good reason. The only thing that improper lists are truly good for is as part of the syntax for association lists—and even there, a custom syntax for key-value pairs would be better. Any use you can think of for improper lists can be better implemented with record types—which, after all, subsume lists: you can define Lisp lists in terms of records, but not the other way around (because lists don't allow you define data structures whose type is disjoint from all other types in the language).
The abuse of pairs and lists to represent all types of data is what I like to call Lisp programmer's disease, and it's a real shame that so many Lisp advocates advocate for it. I've had to clean up that stuff way too many times.