  In this paper, a family of non-weight modules over Lie superalgebras $S(q)$
of Block type are studied. Free $U(\eta)$-modules of rank $1$ over Ramond-Block
algebras and free $U(\mathfrak{h})$-modules of rank $2$ over
Neveu-Schwarz-Block algebras are constructed and classified. Moreover, the
sufficient and necessary conditions for such modules to be simple are
presented, and their isomorphism classes are also determined. The results cover
some existing results.
