I haven't read any more than a few papers on it, and maybe only one of those in depth (which I'll mention below). Mostly I'm going by forum threads, wiki articles, and the design choices certain languages make (like Inform's multimethods and Haskell's type classes).
As far as I understand the history, Philip Wadler's work basically defined the strict parameters of the expression problem and explored solutions for it. Separate compilation and the avoidance of dynamic casts were big deals for Wadler for some reason.
That work was focused on Java, where it's easy to define new classes that implement existing interfaces but impossible to implement new interfaces on existing classes.
The solution I'm most familiar with for Java-style languages is the use of object algebras, as described in Oliveira and Cook's "Extensibility for the Masses: Practical Extensibility with Object Algebras" (https://www.cs.utexas.edu/~wcook/Drafts/2012/ecoop2012.pdf). In this approach, when you extend the system with a new type, you define a new interface with a generic type parameter and a factory method for building that type, and you have that interface inherit all the existing factory methods. So you don't have to solve the unsolvable task of implementing a new interface for an existing class, because you're representing your types as type parameters and methods, not simply as classes.
So I think the main subject of research was how best to represent an extensible program's types and functions in a language like Java where the most obvious choices weren't expressive enough. I think it's more of a "how do we allow extensions to be made at all" problem than a "how do we make all the extensions maintainable" problem.
But then, I've really barely scratched the surface of the research, so I could easily be missing stuff like that.