Structure formation in cosmologies with oscillating dark energy

{abridged} We study the imprints on the formation and evolution of cosmic structures of dynamical dark energy models, characterized by an oscillating equation of state. 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. We consider six different oscillating dark energy models, each characterized by a different set of parameter values. 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. We then estimate the behavior of several cosmological observables, such as the linear growth factor, the Integrated Sachs-Wolfe (ISW) 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 show an oscillating behavior 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. We also notice that deviations with respect to the expectations for a fiducial LambdaCDM 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 \sim 10% over a wide range of angular scales, and the abundance of galaxy clusters, which is modified at the $\sim 10-20%$ level at $z \gtrsim 0.6$ for future wide weak lensing surveys.