We will discuss various techniques that have been introduced to establish non-trivial dynamical properties of solutions to the 2d Euler equation, particularly infinite-time singularity formation. The first lecture will be devoted to basic examples of Arnold's stability theorems as well as proofs of unbounded gradient growth in the 2d Euler equation. The second lecture will be devoted to a different type of argument for unbounded gradient growth of the vorticity based on establishing unbounded gradient growth of the trajectory map along with a Baire-Category argument. The third lecture will be devoted to a recent result on the stability of shearing and various applications. Some papers that might be helpful: 1. S. Denisov, Infinite superlinear growth of the gradient for the two-dimensional Euler equation 2. T. Drivas and T. Elgindi, Singularity formation in the incompressible Euler equation in finite and infinite time. 3. T. Drivas, T. Elgindi, and I. Jeong, Twisting in Hamiltonian Flows and Perfect Fluids 4. A. Kiselev, V. Sverak, Small scale creation for solutions of the incompressible two dimensional Euler equation 5. H. Koch, Transport and Instability for Perfect fluids 6. N. Nadirashvili, Wandering solutions of Euler's D-2 equation