Start up of plane Couette flow and large amplitude oscillatory shear flow of single and multimode Maxwell fluids as well as Oldroyd-B fluids have been analyzed by analytical or semi-analytical procedures. The result of our analysis indicates that if a single or a multimode Maxwell fluid has a relaxation time comparable or smaller than the rate of change of force imparted on the fluid, then the fluid response is not singular as Elasticity Number (E --> infinity). However, if this is not the case, as E --> infinity, perturbations of single and multimode Maxwell fluids give rise to highly oscillatory velocity and stress fields. Hence, their behavior is singular in this limit. Moreover, we have observed that transients in velocity and stresses that are caused by propagation of shear waves in Maxwell fluids are damped much more quickly in the presence of faster and faster relaxing modes. In addition, we have shown that the Oldroyd-B model gives rise to results quantitatively similar to multimode Maxwell fluids at times larger than the fastest relaxation time of the multimode Maxwell fluid. This suggests that the effect of fast relaxing modes is equivalent to viscous effects at times larger than the fastest relaxation time of the fluid. Moreover, the analysis of shear wave propagation in multimode Maxwell fluids clearly show that the dynamics of wave propagation are governed by an effective relaxation and viscosity spectra. Finally, no quasi-periodic or chaotic flows were observed as a result of interaction of shear waves in large amplitude oscillatory shear flows for any combination of frequency and amplitudes.