Groupoids provide a rich supply of C$^{*}$-algebras, and there are many results describing the structure of these C$^{*}$-algebras using properties of the underlying groupoid. For non-Hausdorff groupoids, less is known, largely due to the existence of 'singular' functions in the reduced C$^{*}$-algebra. This talk will discuss how structural properties of the singular ideal can be understood by restricting to isotropy groups. In particular, for amenable second-countable étale groupoids, this technique allows us to characterise when the singular ideal has dense intersection with the underlying groupoid $*$-algebra. This is based on joint work with J. B. Hume.