This project is set in the blooming field of singular stochastic partial differential equations (singular SPDEs) that has undergone a revolution two-three years ago, with the joint introduction by Hairer and Gubinelli-Imkeller-Perkowski of entirely new methods. These works have opened a whole field and offer now the possibility to investigate a number of problems that were out of reach so far. We aim in this project to push further the study of their theoretical foundations and to investigate a number of challenging open questions about some special singular equations. 1. Developing the theory. The theory of regularity structures offers a very clean setting for the study of a class of parabolic singular SPDEs, called sub-critical. So far, the probabilistic structure has mainly been used as a tool to set the study of such an equation into the framework of regularity structures, by enriching the noise into a model. It is very likely that the stochastic cancellations inherent to the probabilistic objects will be instrumental in analyzing a number of problems beyond the local well-posedness problem, such as the global well-posedness problem, or the use of Malliavin calculus tools to investigate the existence and regularity of densities for solutions of some classes of singular sub-critical SPDEs. On the paracontrolled side, the theory was originally written as a 'first order Taylor expansion' theory. Despite its successes in recovering a number of results obtained via the theory of regularity structures, its present form prevents a priori its use in a number of problems and hides one of Hairer's other breakthrough, which is the introduction in his theory of a renormalization group. We intend to develop a higher order paracontrolled calculus and introduce in this setting an analogue of the renormalization group. 2. Qualitative properties of particular singular SPDEs. The powerful tools of regularity structures and paracontrolled calculus have mainly been used so far to derive existence, uniqueness and regularity results for singular PDEs. We feel this is the right time to explore further the full power of these techniques, and other techniques, to get a number of qualitative properties of solutions of important examples of singular equations, such as intermittency for the parabolic Anderson model equation, localization for the Anderson Hamiltonian, the study of the KPZ equation in space dimension greater than 1, or the stochastic Yang-Mills equation that comes from Parisi-Wu Euclidean quantization scheme. At the same time, we shall also investigate a number of PDEs for which totally different tools need to be used, such as fully nonlinear parbolic equations.