Are there properties of rings of which one does not know whether they are Morita or derived invariances?
For a recent such example for Morita invariance, see https://www.sciencedirect.com/science/article/pii/S0021869314004669 .
One can restrict also to special classes of rings such as finite dimensional algebras.
For derived equivalences it is for example an open problem whether having infinite dominant dimension is a derived invariant, see https://link.springer.com/article/10.1007/s11856-016-1327-4 .
