In this talk we describe recent results on classification and rigidity properties of stationary homogeneous solutions to the 3D and 2D Euler equations. The problem is motivated by recent exclusions of self-similar blowup for Euler and its relation to the Onsager conjecture. In 2D the problem also arises in several other areas such as isometric immersions and optimal transport. A full classification of two dimensional solutions will be given. In 3D we reveal several new classes of solutions and prove their rigidity properties. In particular, irrotational solutions are characterized by vanishing of the Bernoulli function; and tangential flows are necessarily 2D axisymmetric pure rotations. In several cases solutions are excluded altogether. The arguments reveal geodesic features of the Euler equation on the sphere. We further discuss the case when homogeneity corresponds to the Onsager-critical state. We will show that anomalous energy flux at the singularity vanishes, which is suggestive of absence of extreme 0-dimensional intermittencies.