Weak uniqueness by noise for singular stochastic PDEs

We prove weak uniqueness of mild solutions for general classes of SPDEs on a Hilbert space. The main novelty is that the drift is only defined on a Sobolev-type subspace and no Hölder-continuity assumptions are required. This framework turns out to be effective to achieve novel uniqueness results for several specific examples. Such wide range of applications is obtained by exploiting either coloured or rougher-than-cylindrical noises.