This thesis explores a range of stability techniques applied to fluid structures that develop in various constant density flows. In particular, the stability of nonlinear structures which develop in rotating plane Couette flow is analyzed using Floquet theory, which allows the global stability of an important secondary nonlinear structure called a Taylor vortex to be determined. From this the distinct tertiary states which emerge as Taylor vortices break down are characterized and their bifurcation behaviour is studied. Also, non-modal stability analyses are conducted in rotating plane Couette flow and annular Poiseuille-Couette flow. In each case the growth mechanisms and the form of the perturbations responsible for the maximum linear energy amplification are discussed. Finally, the non-modal behaviour of the Papkovitch-Fadle operator is treated and its relevance to spatially developing disturbances in Stokes channel flow is examined. The mechanisms and the rates of convergence of the linear spatial energy amplification are investigated and contrasted with temporal energy amplification.