In this talk, I will present the construction of self-similar solutions to the two-dimensional incompressible Euler equations. Extending the works of Elling and Shao-Wei-Zhang, I remove the assumption of rotational symmetry, thereby enlarging the class of self-similar spiral solutions. A key step is the analysis of the linearized operator around the radial solution, and I will discuss both the case when this operator is injective and when it admits a finite-dimensional kernel. Lastly, I will briefly comment on how these self-similar solutions connect to the question of non-uniqueness for the Euler equations.