For some quantitative embedding problems in symplectic geometry, one can reduce the main difficulty to classifying certain singular algebraic curves. In the case of 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. 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 worked out. This is the first step in a program to classify the so-called sesquicuspidal curves, which conjecturally give complete obstructions to stabilized ellipsoid embeddings. This is based on joint work in progress with D. McDuff.