We deal with well-posedness and asymptotic dynamics of a class of coupled systems consisting of linearized 3D Navier–Stokes equations in a bounded domain and a classical (nonlinear) elastic plate/shell equation. We consider three models for plate/shell oscillations: (a) the model which accounts for transversal displacement of a flexible flat part of the boundary only, (b) the model for in-plane motions of a flexible flat part of the boundary, (c) the model which accounts for both transversal and longitudinal displacements. For all three cases we present well-posedness results and prove existence of a compact global attractor. In the first two cases the attractor is of finite dimension and possesses additional smoothness. We do not assume any kind of mechanical damping in the plate component in the case of models (a) and (b). Thus our results means that dissipation of the energy in the fluid due to viscosity is sufficient to stabilize the system in the latter cases.