  {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.
