We consider the classical initial and boundary value problem for the Cahn–Hilliard equation with non-degenerate mobility and singular (e.g., logarithmic) potential. We show that any weak solution converges to a single equilibrium using only minimal assumptions, that is, the existence of a global weak solution which satisfies an energy inequality. This result also holds in the three-dimensional case, which was an open problem so far due to the lack of regularity of solutions, especially when the mobility is just a continuous function and/or the boundary is not too smooth. The result also holds for other models like, for instance, Cahn–Hilliard-Navier–Stokes type systems with unmatched densities and viscosities like the one proposed by Abels, Garcke, and Gr¨un (Math. Models Methods Appl. Sci. 22, 2012).