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We present a numerical study of non-colloidal spherical and rigid particles suspended in Newtonian, shear thinning and shear thickening fluids employing an immersed boundary method. We consider a linear Couette configuration to explore a wide range of solid volume fractions ( $$0.1\leqslant \unicode[STIX]{x1D6F7}\leqslant 0.4$$ ) and particle Reynolds numbers ( $$0.1\leqslant Re_{p}\leqslant 10$$ ). We report the distribution of solid and fluid phase velocity and solid volume fraction and show that close to the boundaries inertial effects result in a significant slip velocity between the solid and fluid phase. The local solid volume fraction profiles indicate particle layering close to the walls, which increases with the nominal $$\unicode[STIX]{x1D6F7}$$ . This feature is associated with the confinement effects. We calculate the probability density function of local strain rates and compare the latter’s mean value with the values estimated from the homogenisation theory of Chateau et al. ( J. Rheol. , vol. 52, 2008, pp. 489–506), indicating a reasonable agreement in the Stokesian regime. Both the mean value and standard deviation of the local strain rates increase primarily with the solid volume fraction and secondarily with the $$Re_{p}$$ . The wide spectrum of the local shear rate and its dependency on $$\unicode[STIX]{x1D6F7}$$ and $$Re_{p}$$ point to the deficiencies of the mean value of the local shear rates in estimating the rheology of these non-colloidal complex suspensions. Finally, we show that in the presence of inertia, the effective viscosity of these non-colloidal suspensions deviates from that of Stokesian suspensions. We discuss how inertia affects the microstructure and provide a scaling argument to give a closure for the suspension shear stress for both Newtonian and power-law suspending fluids. The stress closure is valid for moderate particle Reynolds numbers, $$O(Re_{p})\sim 10$$ .