I don't get a grip of what that exactly means. What IS an abstract singlet, doublet,... under $SU(N)$ or other groups?
"Singlet under $SU(N)$" means that the related representation is invariant under $SU(N)$.
"Doublet" is related in general to $SU(2)$, and corresponds to the fundamental ($2$- dimensional) representation of $SU(2)$
For instance, looking at the "weak interaction", left-handed particles transform as a doublet under $SU(2)_L$ , while right-handed particles transform as a singlet under $SU(2)_L$