When the regularity of a solution to the Euler and Navier-Stokes equation is higher than 1/3 (in a certain Besov sense), then the solution is known to conserve energy.  This threshold is also dimension-independent. At the same time for 3D Euler the convex integration method provided examples of solutions in regularity 1/3 – epsilon that do not conserve energy (P. Isett, 2016).  Finding examples that belong to the critical regularity class 1/3 constitutes what is called the Onsager conjecture.  In this talk we will highlight several special mechanisms that allow to prove energy conservation for incompressible fluids that fall under the Onsager threshold. In particular, this provides a list of scenarios one would like to avoid when attempting to construct critical energy-dissipative examples.