In this talk we will consider geometric properties for solutions to the density-dependent incompressible Euler system in any dimension $N \geq 2$. In particular, we will focus on striated and conormal regularity, which are a natural way of generalizing the classical $2$-D vortex patches structure for homogeneous fluids. We will show that striated (or conormal) regularity for initial density and vorticity is preserved during the time evolution, and it propagates also to velocity field and pressure term. In contrast with the classical case, if the initial vorticity has compact support, this property doesn't propagate, due to the presence of the density term in the vorticity equation (this is also the reason why the results we get are only local in time). Nevertheless, in the physical situation of dimension $N = 2$ or $3$ we will show that, if we focus our attention on the dynamics in the interior of a bounded domain in which initial data are Hölder continuous, then the regularity of solutions is preserved in the domain transported by the flow.