In certain kinematic and particle mass configurations, triangle singularities may lead to line-shapes which mimic the effects of resonances. This well-known effect is scrutinized here in the presence of final-state rescattering. The goal is achieved first by utilizing general arguments provided by the Landau equations, and second by applying a modern scattering formalism with explicit two- and three-body unitarity. In the latter, the effects are studied for a toy-system which resembles the so-called $a_1(1420)$-meson.