I would be very interested to see a single real-world use for a fixed-point combinator. I have a reasonable understanding of what one does, but I still haven't got the faintest idea why you would ever need one, nor have I seen a single situation in which applying one doesn't make the situation unimaginably more complicated to understand.
Fixed point operators are heavily used in abstract interpretation (https://secure.wikimedia.org/wikipedia/en/wiki/Abstract_inte...), which in turn is used in static analysis of programs for automatic bug detection or compiler optimizations. In certain situations it can also be used to formally prove the correctness of a program. Just to give a concrete example, this is a static analyzer based on abstract interpretation: http://www.astree.ens.fr/ . Polyspace (https://secure.wikimedia.org/wikipedia/en/wiki/Polyspace) is another static analysis tool that uses abstract interpretation.
I think Haskell uses a fixpoint combinator built into the core language under some syntactic sugar to enable any sort of recursion, so the real-world use would be pretty much making every Haskell program ever work.
