For some quantitative embedding problems in symplectic geometry, one can reduce the main difficulty to classifying certain algebraic singular curves. For the complex projective plane these curves are now fairly well-understood, but even for other simple del Pezzo surfaces there is a whole complicated zoo of such curves and there is not yet any suitable classification result. In this talk, I will explain how the formalism of cluster algebras gives a beautiful way of systematically organizing and understanding unicuspidal curves. In particular, they are organized into the exchange graph of an associated cluster algebra, and in some cases these exchange graphs have been fully classified. I will not assume any prior familiarity with cluster algebras. This is based on joint work in progress with D. McDuff.