  Let $H$ be a numerical semigroup. We give effective bounds for the
multiplicity $e(H)$ when the associated graded ring
$\operatorname{gr}_\mathfrak{m} K[H]$ is defined by quadrics. We classify
Koszul complete intersection semigroups in terms of gluings. Furthermore, for
several classes of numerical semigroups considered in the literature
(arithmetic, compound, special almost complete intersections, $3$-semigroups,
symmetric or pseudo-symmetric $4$-semigroups) we classify those which are
Koszul.
