  We investigate lineability/spaceability problems within the setting of
multilinear summing operators on quasi-Banach sequence spaces. Furthermore, we
deal with the contemporary geometric notions of pointwise-lineability and
$(\alpha,\beta)$-lineability. Among other results, we prove that the class of
multilinear operators taking values on $\ell_q \, (0<q \leq \infty)$ that are
absolutely but not multiple summing is pointwise $\mathfrak{c}$-spaceable when
non-empty. Spaceability in other classes, such as Dunford-Pettis operators and
multilinear operators on general summable scalar families, is also studied.
