2-semiarcs in $\mathrm{P}\mathrm{G}(2,q)$, $q\le 13$

A $2$-semiarc is a pointset ${\mathcal{S}}_k$ with the property that the number of tangent lines to ${\mathcal{S}}_k$ at each of its points is two. Using some theoretical results and computer aided search, the complete classification of $2$-semiarcs in PG$(2,q)$ is given for $q\le 7,$ the spectrum of their sizes is determined for $q\le 9$, and some results about the existence are proven for $q=11$ and $q=13.$ For several sizes of $2$-semiarcs in $\mathrm{P}\mathrm{G}(2,q)$, $q\le 7$, classification results have been obtained by theoretical proofs.