Weak solutions arise naturally in the study of the Navier-Stokes and Euler equations both from an abstract regularity/blow-up perspective and from physical theories of turbulence. This thesis studies the structure and size of singular set of such weak solutions to equations of incompressible fluid dynamics from two opposite directions. First, it aims to single-out new mechanisms which allow to break the typically supercritical scaling of the equations and, in this way, prevent the formation of singularities either globally or locally in spacetime. Second, in the absence of such mechanisms, we seek to quantify how singular (in terms of dimension of the singular set, for instance) the solutions that we are actually able to construct are.

This thesis collects four results pointing in the two directions outlined above which have been obtained in several collaborations during the Ph.D. studies:
- a global regularity result for the fractional Navier-Stokes equation slightly blow the critical fractional order,
- a global well-posedness result for the defocusing wave equation with slightly supercritical power nonlinearity,
- an a.e. smoothness / partial regularity result for the supercritical surface quasigeostrophic (SQG) equation,
- an estimate (and a discussion of its sharpness) on the dimension of the singular set of wild Hölder continuous solutions of the incompressible Euler equations.

All results presented in the thesis have either been published or are submitted for publication.