  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 $q\leq 7,$ the spectrum of their
sizes is determined for $q\leq 9$, and some results about the existence are
proven 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 by
theoretical proofs.
