The stability of buoyancy-driven parallel shear flow of a couple stress fluid confined between vertical plates is investigated by performing a classical linear stability analysis. The plates are maintained at constant but different temperatures. A modified Orr-Sommerfeld equation is derived and solved numerically using the Galerkin method with wave speed as the eigenvalue. The critical Grashof number G(c), critical wave number a(c) and critical wave speed c(c) are computed for wide ranges of couple stress parameter A(c) and the Prandtl number Pr. Based on these parameters, the stability characteristics of the system are discussed in detail. The value of Prandtl number, at which the transition from stationary to travelling-wave mode takes place, increases with increasing A(c). The couple stress parameter shows destabilising effect on the convective flow against stationary mode, while it exhibits a dual behaviour if the instability is via travelling-wave mode. The streamlines and isotherms presented demonstrate the development of complex dynamics at the critical state. (C) 2014 Elsevier Ltd. All rights reserved.