This study investigates the oscillation behavior of fluidic oscillators when operating with non-Newtonian power-law fluids, focusing on how rheological parameters and chamber geometry influence oscillation onset and stability. Using unsteady numerical simulations, the researchers analyze the coupling effects of the flow behavior index (n), which reflects shear-thinning capability, and the consistency index (K), which measures overall viscous resistance. They identify a critical threshold (n≈0.61, K≈0.163) beyond which oscillations fail due to excessive viscous resistance, disrupting the Coanda effect. The study distinguishes two oscillation cessation states: marginal locking, where increased inlet velocity can reactivate oscillation, and deep locking, where high consistency creates irreversible deadlock. Additionally, the chamber expansion angle (α) significantly impacts stability, with angles above 20° weakening wall-attachment pressure gradients even in strongly shear-thinning fluids. These findings establish a coupled rheology-geometry-dynamics criterion for oscillation onset, offering valuable guidance for designing fluidic components and optimizing performance when working with complex fluids.