Context: The peculiar three-peak structure of the linear polarization profile shown in the second solar spectrum by the Ba ii line at 4554 Å has been interpreted as the result of the different contributions coming from the barium isotopes with and without hyperfine structure. In the same spectrum, a triple peak polarization signal is also observed in the Sc ii line at 4247 Å. Scandium has a single stable isotope (45Sc), which shows hyperfine structure due to a nuclear spin I = 7/2. Aims: We investigate the possibility of interpreting the linear polarization profile shown in the second solar spectrum by this Sc ii line in terms of hyperfine structure. Methods: A two-level model atom with hyperfine structure is assumed. Adopting an optically thin slab model, the role of atomic polarization and of hyperfine structure is investigated, avoiding the complications caused by radiative transfer effects. The slab is assumed to be illuminated from below by the photospheric continuum, and the polarization of the radiation scattered at 90° is investigated. Results: The three-peak structure of the scattering polarization profile observed in this Sc ii line cannot be fully explained in terms of hyperfine structure. Conclusions: Given the similarities between the Sc ii line at 4247 Å and the Ba ii line at 4554 Å, it is not clear why, within the same modeling assumptions, only the three-peak Q/I profile of the barium line can be fully interpreted in terms of hyperfine structure. The failure to interpret this Sc ii polarization signal raises important questions, whose resolution might lead to significant improvements in our understanding of the second solar spectrum. In particular, if the three-peak structure of the Sc ii signal is actually produced by a physical mechanism neglected within the approach considered here, it will be extremely interesting not only to identify this mechanism, but also to understand why it seems to be less important in the case of the barium line.