In this talk, we will discuss the interaction between the stability, and the propagation of regularity, for solutions to the incompressible 3D Euler equation. It is still unknown whether a solution with smooth initial data can develop a singularity in finite time. We will explain why the prediction of such a blow-up, via direct numerical experiments, is so difficult. We will describe how, in such a scenario, the solution becomes unstable as time approaches the blow-up time. The method use the relation between the vorticity of the solution, and the bi-characteristic amplitude solutions, which describe the evolution of the linearized Euler equation at high frequency. In the axisymmetric case, we can also study the instability of blow-up profiles. This work was partially supported by the NSFDMS-1907981.