Summary: Rather general sufficient conditions are found for a given multivalued mapping \(F : X \to Y\) to possess an upper semicontinuous and compact-valued selection \(G\) which is defined on a dense \(G_\delta\)-subset of the domain of \(F\). The case when the selection \(G\) is single-valued (and continuous) is also investigated. The results are applied to prove some known as well as new results concerning generic differentiability of convex functions, Lavrentieff type theorem, generic well-posedness of optimization problems and generic nonmultivaluedness of metric projections and antiprojections.