This study examines the impact of dilute polymer solutions on the unique isolated secondary flows between concentric, rotating cylinders, namely Taylor-Couette (TC) flow. We mapped the stability of flow states using Newtonian and dilute polyethylene oxide (PEO) solutions over the Reynolds number range of - 100 < Re-o < 500 and 0 < Re-i < O(10(3)), where subscripts 'o' and 'i' refer to outer and inner cylinders, respectively. Elasticity number (El) of the PEO fluids, defined as the ratio of elastic to inertial forces, ranges from O(10(-4)) to O(10(-2)). This work expands on previous studies by (a) significantly expanding the range of Re-i, Re-o, and El examined, (b) use of a consistent, conservative protocol for reaching flow states, and (c) rheological characterization of the solutions via shear and capillary breakup extensional rheometry. Using spectral analysis of flow visualization of the r-z or z-theta planes, we find the effect of El on the critical conditions for laminar and chaotic axisymmetric and nonaxisymmetric flow states is nonmonotonic and mode-dependent, with greater modification of higher order transitions involving small-scale features. While the critical conditions are modified by low El for all transitions, the flow states vary from those for Newtonian fluids at higher Re-i and for the more elastic fluids. (C) 2011 The Society of Rheology. [DOI: 10.1122/1.3626584]