Algebraic singular functions are not always dense in the ideal of $C^{\ast }$-singular functions

We give the first examples of étale (non-Hausdorff) groupoids $\mathcal{G}$ whose $C^{\ast }$-algebras contain singular elements that cannot be approximated by singular elements in ${\mathcal{C}}_c(\mathcal{G})$. 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 $C^{\ast }$-algebra.