Abstract We initiate a detailed analysis of $C^{\ast }$-diagonals in classifiable }$-algebras, answering natural questions concerning topological properties their spectra and uniqueness questions. Firstly, we construct with connected all stably finite which are unital or projectionless continuous scale. Secondly, for }$-algebras torsion-free $K_0$ trivial $K_1$, further determine the up to home...