Their theorem is that for every compact 3-dimensional manifold $M$ in the space $Riem(M)$ (consisting of all Riemannian metrics on $M$ equipped with $C^\infty$ topology), there exists a countable union of nowhere dense closed subsets: $$ C= \bigcup_{n\in {\mathbb N}} C_n\subset Riem(M) $$ such that for every metric $g\in U:=Riem(M) \setminus C$, Yau's conjecture holds. The set $U$ can be regarded as a set of "generic" Riemannian metrics on $M$. (For instance, it is dense in $Riem(M)$.)
As an aside, their proof depends on the earlier result by Liokumovich, Marques and Neves proving a "nonlinear version of Weyl's law" (conjectured by Gromov) for minimal surfaces.
I am not sure though how much the terminology I used makes sense to you.