After reading sbi and Eli Bendersky's answers in this question I started to wondering what static member functions are for.
A class' friend free function shouldn't be able to do anything a static member function can do? If so, why/when should I prefer a static member function to a friend free one?
In general:
Require access to private members
static member functions have access to private members of the class. If you need that, you can use a static member function. You have to declare it in the header anyway to give it access, so you may as well make it a member rather than a friend. It is commonly done this way for singletons that have a getInstance() method as singleton, and classes that use a static factory method createInstance() to ensure they are created on the heap. Both of these need access to the private constructor.
Meta-programming
static member functions are very good for template meta-programming where you can pass in a class and call its method without knowing at the point of call what function will actually get invoked. This is commonly called "compile-time polymorphism" and is an important part of meta-programming. std::char_traits is based on this principle.
Restricted access
The common use of a private static member function, just so that it can be accessed only by the class, and does not itself need access to private members, is not a good use of static member functions because it is part of the implementation detail of the class, and this is better done in the anonymous namespace of the compilation unit.
However if the static member function is protected it has use as it can get called by derived classes but not by external classes.
friend functions
operator<<
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