Do singular distributions have any real-world applications? Do singular probability distributions have any real-world applications, or are they just a pure-mathematical curiosity? I can't imagine a real quantity that they would describe. But on the other hand, singular functions do have surprising applications, e.g. regarding the fractional quantum Hall effect in condensed-matter physics, so maybe singular probability distributions do as well.
By "real-world application" I don't mean a real phenomenon which could in principle be modeled by a singular distribution, but rather a situation in which engineers, scientists, financial analysts, or other non-mathematicians actually do use them in the context of a commercial application or other non-mathematical "product".

Here is one application that might interest you. Consider an integrable function f on $\mathbb R$ such that $2f(x)=3f(3x)+3f(3x-1)$ a.e.. It turns out that $f=0$ a.e.. There seems to be no simple way of doing this even if you assume that f is smooth function. There is an elegant proof by relating it to cantor set and a singular distribution function. This is Problem 261 in 'Exercises in Analysis 201-300' at statmathbc.wordpress.com where complete solutions are provided on request.
