Do all distributions of R.V.s have a singular part and a continuous part? Consider the probability distribution of a real-valued R.V. as the equivalence class of generalized PDFs where the integral over each measurable set in $\mathbb{R}$ is the same in each PDF. 
1) Can any R.V.'s distribution be represented as the sum of a normal function and a countable number of $\delta$'s? 
2) If so, does there exist an element in the equivalence class where the 'normal function' part's set of discontinuities is nowhere dense?
I am trying to work out the form of a general product distribution and it would be helpful to know what distributions look like.

There exist distributions which are neither discrete nor continuous.  For example, the Lebesgue-Stieltjes measure generated by the Cantor function.
