  Effective string theory has shown its universal power in the prediction of
the spectrum of low-lying excited states of confining strings. Here we study
confining flux tubes in $\mathbb{Z}_N$ gauge theories. For the $N=2$ theory,
which corresponds to the 3d Ising gauge model, we compute the spectrum of
low-lying excitations of confining strings and show that it agrees with the
universal Nambu--Goto predictions except for an additional massive scalar
resonance. This resonance, however, turns out to be a bulk glueball mixing with
the flux tube excitations rather than a genuine string worldsheet state. In
general $\mathbb{Z}_N$ gauge theories (dual to clock spin models), we observe a
continuous phase transition for $N \geq 4$, while for $N > 5$ it is governed by
the $O(2)$ universality class. The critical behavior of the string tension and
mass gap is verified to be described by a dangerously irrelevant operator. At
large $N$ the glueball spectrum is expected to approach the spectrum of U(1)
gauge theory, which is confirmed by our lattice data.
