We study the flow stability and spatiotemporal spectral dynamics of cellulose nanocrystal (CNC) suspensions in a custom Taylor-Couette flow cell using the intrinsic shear induced birefringence and liquid crystalline properties of CNC suspensions for flow visualizations, for the first time. The analysis is performed at constant ramped speed inputs of the independently rotating cylinders for several cases ranging from only inner or outer rotating cylinders to three counter-rotation cases. All CNC suspensions have measurable elasticity and shear thinning, both increasing with CNC concentration. We show that the flow patterns recorded are essentially Newtonian-like, with non-Newtonian effects ranging from a decrease in wavenumbers to altering the critical parameters for the onset of instability modes. Outer cylinder rotation flow cases are stable for all concentrations whereas inner cylinder rotation flow cases transition to axisymmetric and azimuthally periodic secondary flows. However, counter-rotation cases become unstable to asymmetric spiral modes. With increasing CNC concentration, a counter-rotation case was found where azimuthally periodic wavy patterns transition to asymmetric spiral modes. Based on rheo-SAXS measurements, the shear-thinning region of CNC suspensions is expected to lead to the breakdown of the chiral nematic phase, whose elastic constants constitute the dominant structural elasticity mechanism. Thus, we interpret the Taylor-Couette stability of the CNC suspensions as dominated by their shear-thinning character due to the expected loss of elasticity in nonlinear flow conditions.