The purpose of this paper is to examine whether the presence of singular configurations poses an obstruction to the small-time local controllability (STLC) and stabilizability of the attitude dynamics of a spacecraft actuated by single-gimbal control moment gyroscopes (CMGs). We identify a class of singular CMG configurations called critically singular configurations, at which the linearized dynamics fail to be controllable. Linear controllability, and hence STLC and stabilizability, hold at equilibria in which the CMG array is in a nonsingular or singular, but not critically singular, configuration. We give sufficient conditions and necessary conditions on critically singular configurations for STLC to hold. In particular, we show that STLC and stabilizability fail to hold at an equilibrium if the magnitude of the total spin angular momentum of the CMG array has a local maximum at the equilibrium.