The instability mechanism of the shear-thinning and shear-thickening fluids pasta circular cylinder is studied using linear theory. The shear-dependent viscosityis modeled by the Carreau-law where the rheological parameters, the power-index and the material time constant are chosen in the range 0.4 < n < 1.75 and0.1 < λ < 100. A second order finite-difference code is used for the simulationof cylinder flow in which the Immersed Boundary Technique is implemented torepresent the cylinder surface on a Cartesian mesh. The critical Reynolds num-ber for the onset of instability is reported for a range of rheological parameters.Structural sensitivity analysis based on the idea of ”wavemaker” is performedto identify the core of the instability. Perturbation kinetic energy budget is alsoconsidered to examine the physical mechanism of the instability. The charac-teristics of base-flow: drag coefficient, size of recirculation bubble and viscositydistribution are presented to provide useful knowledge about shear-thinning ef-fect in flow past a cylinder.