In the 1960s, Robert Kraichnan introduced an idealized model for passive scalar turbulence, in which the scalar field is advected by a Gaussian velocity that is delta-correlated in time and Hölder continuous in space. Despite its simplicity, the corresponding linear stochastic PDE captures key features of turbulent flows, including anomalous dissipation. Renewed interest in this model followed the work of Coghi and Maurelli, which showed that the same transport-type noise restores well-posedness in regimes where the deterministic 2D Euler equations admit non-unique weak solutions. In this talk, we further develop this line of research by investigating additional properties of the solutions constructed by Coghi and Maurelli. In particular, we present new results on anomalous fractional Sobolev regularity and anomalous dissipation of the mean enstrophy for solutions to the 2D Euler equations with rough Kraichnan noise. Time permitting, we will also discuss implications for the well-posedness theory of more singular nonlinear advection models, such as the Surface Quasi-Geostrophic and Incompressible Porous Media equations. This talk is based on ongoing joint work with L. Galeati and U. Pappalettera.