We study the imprints on the formation and evolution of cosmic structures of a particular class of dynamical dark energy models, characterized by an oscillating equation of state. This investigation complements earlier work on the topic that focused exclusively on the expansion history of the Universe for such models. Oscillating dark energy cosmologies were introduced in an attempt to solve the coincidence problem, since in the course of cosmic history matter and dark energy would have had periodically comparable energy densities. In this class of models the redshift evolution of the equation of state parameter w(z) for dark energy is characterized by two parameters, describing the amplitude and the frequency of the oscillations (the phase is usually set by the boundary condition that w(z) should be close to −1 at recent times). We consider six different oscillating dark energy models, each characterized by a different set of parameter values. For one of these models w(z) is lower than −1 at present and larger than −1 in the past, in agreement with some marginal evidence from recent Type Ia supernova studies. Under the common assumption that dark energy is not clustering on the scales of interest, we study different aspects of cosmic structure formation. In particular, we self-consistently solve the spherical collapse problem based on the Newtonian hydrodynamical approach, and compute the resulting spherical overdensity as a function of cosmic time. We then estimate the behaviour of several cosmological observables, such as the linear growth factor, the integrated Sachs–Wolfe effect, the number counts of massive structures and the matter and cosmic shear power spectra. We show that, independently of the amplitude and the frequency of the dark energy oscillations, none of the aforementioned observables shows an oscillating behaviour as a function of redshift. This is a consequence of the said observables’ being integrals over some functions of the expansion rate over cosmic history, thus smoothing any oscillatory features in w(z) below detectability. We also notice that deviations with respect to the expectations for a fiducial Λ cold dark matter cosmology are generically small, and in the majority of the cases distinguishing an oscillating dark energy model would be difficult. Exceptions to this conclusion are provided by the cosmic shear power spectrum, which for some of the models shows a difference at the level of ∼10 per cent over a wide range of angular scales, and the abundance of galaxy clusters, which is modified at the ∼10–20 per cent level at z≳ 0.6 for future wide weak lensing surveys.