Understanding the nature of Dark Energy (DE) in the Universe is the central challenge of modern cosmology. Einstein’s Cosmological Constant (Λ) provides the simplest explanation fitting the available cosmological data thus far. However, its unnaturally tuned value indicates that other hypothesis must be explored. Furthermore, current observations do not by any means rule out alternative models in favor of the simplest “concordance” ΛCDM. In the absence of theoretical prejudice, observational tests have mainly focused on the DE equation of state. However, the detection of the inhomogeneous nature of DE will provide smoking-gun evidence that DE is dynamical, ruling out Λ. This key aspect has been mostly overlooked so far, particularly in the optimization design of the next generation of surveys dedicated to DE searches which will map the distribution of matter in the Universe with unprecedented accuracy. The success of these observations relies upon the ability to model the non-linear gravitational processes which affect the collapse of Dark Matter (DM) at small and intermediate scales. Therefore, it is of the highest importance to investigate the role of DE inhomogeneities throughout the non-linear evolution of cosmic structure formation. To achieve this, we will use specifically designed high-resolution numerical simulations and analytical methods to study the non-linear regime in different DE models. The hypothesis to be tested is whether the intrinsic clustering of DE can alter the predictions of the standard ΛCDM model. We will investigate the observational consequences on the DM density field and the properties of DM halos. The results will have a profound impact in the quest for DE and reveal new observable imprints on the distribution of cosmic structures, whose detection may disclose the ultimate origin of the DE phenomenon.