In this study we have theoretically investigated the effect of detailed fluid rheology (e.g., spectrum of relaxation times, shear thinning of first normal stresses, finite second normal stresses, and ratio of solvent to total viscosity) and energetics on the purely elastic instability of Taylor-Couette flow. The isothermal analysis showed that irrespective of the details of the fluid rheology, solvent to total viscosity ratio and gap width, the secondary flow is time-dependent and nonaxisymmetric. However, as the number of relaxation times is increased, the critical Deborah number reaches an asymptotic value which is approximately half of the critical Deborah number predicted by the single-mode constitutive equation. These results strengthen the conclusions of Al-Mubaiyedh et al. (1999) that have attributed the existence of a stationary secondary how in the experiments of Baumert and Muller (1995, 1997) to the effect of energetics. The nonisothermal analyses predicted, for experimentally realizable values of Peclet and Brinkman numbers, the instability to be caused by stationary and axisymmetric disturbances as observed in the experiments of Baumert and Muller (1995, 1997). Moreover, it has been shown that although the details of the fluid rheology can influence the onset conditions in the nonisothermal viscoelastic Taylor-Couette flow, the mechanism of the instability as well as the structure of the bifurcated solutions are relatively insensitive to the details of the fluid rheology.