  Quasi-socle ideals, that is the ideals $I$ of the form $I= Q :
\mathfrak{m}^q$ in Gorenstein numerical semigroup rings over fields are
explored, where $Q$ is a parameter ideal, and $\mathfrak{m}$ is the maximal
ideal in the base local ring, and $q \geq 1$ is an integer. The problems of
when $I$ is integral over $Q$ and of when the associated graded ring
$\mathrm{G}(I) = \bigoplus_{n \geq 0}I^n/I^{n+1}$ of $I$ is Cohen-Macaulay are
studied. The problems are rather wild; examples are given.
