  We construct C*-diagonals with connected spectra in all classifiable stably
finite C*-algebras which are unital or stably projectionless with continuous
scale. For classifiable stably finite C*-algebras with torsion-free $K_0$ and
trivial $K_1$, we further determine the spectra of the C*-diagonals up to
homeomorphism. In the unital case, the underlying space turns out to be the
Menger curve. In the stably projectionless case, the space is obtained by
removing a non-locally-separating copy of the Cantor space from the Menger
curve. We show that each of our classifiable C*-algebras has continuum many
pairwise non-conjugate such Menger manifold C*-diagonals. Along the way, we
also obtain a complete classification of C*-diagonals in all one-dimensional
non-commutative CW complexes.
