I find the question perhaps a little too soft for MO, in that it might be closed as subjective.

One of the key problems with "functions with singularities" is you can usually just change the domain of the function, to get a smooth function on another domain (now with its own singularities, in a sense). This is the kind of thing we're doing (more or less) when we talk about piecewise C^1 or piecewise analytic functions. And these come up naturally enough, for example in a collision between two bodies that involves friction. So although you can approximate by smooth functions, it's in no way "natural" to the model.

Of course there's a very simple answer to your question -- things like probability distributions are by-design a type of function "with singularities".

@dr: Just consider the class of all probability measures on the real numbers. Among which there is one called the Dirac measure which is definitely not regular. So if you study probability, you'd have to (at least be prepared to) deal with singular stuff all the time.

For your second question: consider an arbitrary meromorphic on the complex plane with poles at points z1, z2, ...,

Now consider the same function, but now restricted to C \ {z1,z2,...}. By restricting the domain, the function is now complex analytic. Suddenly you went from having singularities to as smooth as possible.

Consider also Lusin's theorem, which implies the following Corollary: given an absolutely continuous function f defined on the interval [0,1], for every positive epsilon, there is a subset E of measure less than epsilon, such that f is C^1 on [0,1] \ E.