Algebraic singular functions are not always dense in the ideal of C^*-singular functions

We give the first examples of étale (non-Hausdorff) groupoids G whose C^*-algebras contain singular elements that cannot be approximated by singular elements in Cc( G). We provide two examples: one is a bundle of groups, and the other a minimal and effective groupoid constructed from a self-similar action on an infinite alphabet. Moreover, we also prove that the Baum--Connes assembly map for the first example is not surjective, not even on the level of its essential C^*-algebra.