I tentatively suggest that inheritence should usually be reserved for fixing problems in the original design.
Well, the first project in my current job was adding features a semi-complex web application -- written in PHP. The previous developer, no doubt a good programmer otherwise, apparently felt too energetic, i.e. not lazy enough, when he wrote the original source files, because there is considerable overlap in functionality. Quite often this is because he used copy-paste to implement features on pages that lacked them. Needless to say, when I was asked to change the way some summary fields in reports are computed, I had to first manually read through all files and discover the five or six places where the same copy-pasted computation took place.
Now, the job is nice, and I've actually had fun refactoring this. My very first inclination was to abstract the common code to functions, then pass these functions around, akin to higher-level programming. However, although PHP supports lambda expressions through eval, this quickly turned out to be infeasible. Instead, I implemented a couple of shallow class hierarchies and abstracted most common functionality to (abstract) base classes. It's not beautiful or elegant, but it is much cleaner than the original -- plus adding new features is considerably easier now.
Arguably this is not refactoring the design that much; just implementing the design in a bit better way. However, it's a good example where knowing object-oriented programming (inheritance too!) saved the day.