  We completely classify all plane curves of degree at most 30 with a unique
cuspidal (locally unibranch) singular point and rational normalization in terms
of the Newton pairs parameterizing the cusp. We distinguish between prime and
composite degree in the classification and study the relationship between prime
or composite degree and the number of distinct topological types of cuspidal
singularities. Motivated by wall-crossing in moduli of curves, we also survey
several geometric properties of rational unicuspidal plane curves.
