  In this paper we will study the projetivity of various natural modules
associated to operator Segal algebras of the Fourier algebra of a locally
compact group. In particular, we will focus on the question of identifying when
such modules will be projective in the category of operator spaces.
Projectivity often implies that the underlying group is discrete or even
finite. We will also look at the projectivity for modules of $A_{cb}(G)$, the
closure of $A(G)$ in the space of its completely bounded mutipliers. Here we
give an evidence to show that weak amenability of $G$ plays an important role.
