  We introduce a weak solution concept (called "rough weak solutions") for
singular SDEs with additive alpha-stable L\'evy noise (including the Brownian
noise case) and prove its equivalence to martingale solutions from Kremp,
Perkowski '22 in the Young and rough regularity regime. In the rough regime
this leads to the construction of certain rough stochastic sewing integrals
involved. For rough weak solutions, we can then prove a generalized It\^o
formula. Furthermore, we show that canonical weak solutions are wellposed in
the Young case (and equivalent to rough weak solutions), while ill-posed in the
rough case. For the latter, we construct a counterexample for uniqueness in
law.
