We study ω-limit and minimal sets of skew-product semiflows associated with cocycles in Banach spaces, which admit one-dimensional inertial manifolds. Our main aim is to extend for such cocycles classical results of W. Shen and Y. Yi on almost automorphy of minimal sets arising in the case of scalar almost periodic ODEs and scalar almost periodic parabolic equations in one-dimensional domains. We give applications for ODEs, delay equations and semilinear parabolic equations. Conditions for the existence of inertial manifolds are given in our adjacent works. In applications, these conditions are verified with the aid of recently obtained versions of the Frequency Theorem.