  Generalized symmetries often appear in the form of emergent symmetries in low
energy effective descriptions of quantum many-body systems. Non-invertible
symmetries are a particularly exotic class of generalized symmetries, in that
they are implemented by transformations that do not form a group. Such
symmetries appear generically in gapless states of quantum matter, constraining
the low-energy dynamics. To provide a UV-complete description of such
symmetries, it is useful to construct lattice models that respect these
symmetries exactly. In this paper, we discuss two families of one-dimensional
lattice Hamiltonians with finite on-site Hilbert spaces: one with (invertible)
$S^{\,}_3$ symmetry and the other with non-invertible $\mathsf{Rep}(S^{\,}_3)$
symmetry. Our models are largely analytically tractable and demonstrate all
possible spontaneous symmetry breaking patterns of these symmetries. Moreover,
we use numerical techniques to study the nature of continuous phase transitions
between the different symmetry-breaking gapped phases associated with both
symmetries. Both models have self-dual lines, where the models are enriched by
so-called intrinsically non-invertible symmetries generated by
Kramers-Wannier-like duality transformations. We provide explicit lattice
operators that generate these non-invertible self-duality symmetries. We show
that the enhanced symmetry at the self-dual lines is described by a 2+1D
symmetry-topological-order (SymTO) of type $\mathrm{JK}^{\,}_4\boxtimes
\overline{\mathrm{JK}}^{\,}_4$. The condensable algebras of the SymTO determine
the allowed gapped and gapless states of the self-dual $S^{\,}_3$-symmetric and
$\mathsf{Rep}(S^{\,}_3)$-symmetric models.
