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A $2$-semiarc is a pointset ${\mathcal S}_k$ with the property that thenumber of tangent lines to ${\mathcal S}_k$ at each of its points is two. Usingsome theoretical results and computer aided search, the complete classificationof $2$-semiarcs in PG$(2,q)$ is given for $q\leq 7,$ the spectrum of theirsizes is determined for $q\leq 9$, and some results about the existence areproven for $q=11$ and $q=13.$ For several sizes of $2$-semiarcs in$\mathrm{PG}(2,q)$, $q\leq 7$, classification results have been obtained bytheoretical proofs.

2-semiarcs in $\mathrm{PG}(2,q)$, $q\leq 13$