A convenient way to study phase transitions is by considering finite spin systems of linear size
L with boundary conditions that produce a system-size interface. In this paper, we study the statistical properties of such an interface in a disordered Potts ferromagnet in dimension
d=2 within Migdal--Kadanoff real-space renormalization. We first focus on the interface free-energy and energy to measure the singularities of the average and random contributions, as well as the corresponding histograms, both in the low-temperature phase and at criticality. We then consider the critical behavior of the interfacial adsorption of nonboundary states. Our main conclusion is that all singularities involve the correlation length ${\ensuremath{\xi}}_{\mathit{av}}(T)\ensuremath{\sim}{({T}_{c}\ensuremath{-}T)}^{\ensuremath{-}\ensuremath{\nu}}$ appearing in the average free-energy $\overline{F}\ensuremath{\sim}{[L∕{\ensuremath{\xi}}_{\mathit{av}}(T)]}^{{d}_{s}}$ of the interface of dimension ${d}_{s}=d\ensuremath{-}1$, except for the free-energy width $\ensuremath{\Delta}F\ensuremath{\sim}{[L∕{\ensuremath{\xi}}_{\mathit{var}}(T)]}^{\ensuremath{\theta}}$ that involves the droplet exponent $\ensuremath{\theta}$ and another correlation length ${\ensuremath{\xi}}_{\mathit{var}}(T)$ that diverges more rapidly than ${\ensuremath{\xi}}_{\mathit{av}}(T)$. We compare with the spin-glass transition in
d=3, where ${\ensuremath{\xi}}_{\mathit{var}}(T)$ is the ``true'' correlation length and where the interface energy presents unconventional scaling with a chaos critical exponent ${\ensuremath{\zeta}}_{c}>1∕\ensuremath{\nu}$ [M. Nifle and H. J. Hilhorst, Phys. Rev. Lett. 68, 2992 (1992)]. The common feature is that in both cases, the characteristic length scale
Lch(T) associated with the chaotic nature of the low-temperature phase diverges more slowly than the correlation length.