  This work is largely focused on extending D. Higgs' $\Omega$-sets to the
context of quantales, following the broad program of U. H\"ohle, we explore the
rich category of $\mathscr Q$-sets for strong, integral and commutative
quantales, or other similar axioms. The focus of this work is to study the
different notion of 'completeness' a $\mathscr Q$-set may enjoy and their
relations, completion functors, resulting reflective subcategories, their
relations to relational morphisms. We establish the general equivalence of
singleton complete $\mathscr Q$-sets with functional morphisms and the category
of $\mathscr Q$-sets with relational morphisms; we provide two
characterizations of singleton completeness in categorical terms; we show that
the singleton complete categorical inclusion creates limits.
