In the 60’s, Kraichnan proposed a synthetic model for passive scalar turbulence, consisting of a scalar advected by a random Gaussian velocity field, white in time and $\alpha$-Hölder continuous in space. Despite its simplicity, this SPDE displays anomalous dissipation of energy, spontaneous stochasticity and intermittency, which are also expected for more realistic turbulent fluids. At the same time, solutions to the inviscid SPDE are unique and can be recovered by vanishing viscosity and mollification schemes. In this talk I will present some recent further understandings on this model: i) solutions to the transport equation with $L^2$ initial data display anomalous regularisation and almost gain Sobolev regularity $H^{1-\alpha}$, but not better (see [1,2]); ii) solutions to the continuity equation starting from Dirac deltas instantaneously gain Lebesgue integrability, due to the diffusive behaviour of Lagrangian particle splitting, and their variance at small times grows like $t^{1/(1-\alpha)}$ (see [2]).