We discuss the construction of the holonomy groupoid associated with a singular foliation as a quotient space. This groupoid is highly pathological, nevertheless we explain that our construction allows for the generalization of the $C^*$-algebra of the holonomy groupoid to the singular case. We also discuss how this $C^*$-algebra is the recipient for the index of elliptic longitudinal pseudodifferential operators, as well as quantization problems.