We consider SDEs with drift in negative Besov spaces and random initial condition and investigate them from two different viewpoints. In the first part we set up a martingale problem and show its well-posedness. We then prove further properties of the martingale problem, like continuity with respect to the drift and the link with the Fokker-Planck equation. In the second part we show that the solutions are weak Dirichlet processes for which we evaluate the quadratic variation of the martingale component. We then introduce a notion of solution to SDEs with negative Besov drifts, and under suitable assumption we show equivalence with the solution to the martingale problem.