  We consider a class of bulk-surface coupled Cahn-Hilliard systems in a
smooth, bounded domain $\Omega\subset\mathbb{R}^{d}$ $(d\in\{2,3\})$, where the
trace value of the bulk phase variable is connected to the surface phase
variable via a Dirichlet boundary condition or its Robin approximation. For a
general class of singular potentials (including the physically relevant
logarithmic potential), we establish the regularity propagation of global weak
solutions to the initial boundary value problem. In particular, when the
spatial dimension is two, we prove the instantaneous strict separation
property, which ensures that every global weak solution remains uniformly away
from the pure states $\pm 1$ after any given positive time. In the
three-dimensional case, we obtain the eventual strict separation property that
holds for sufficiently large time. This strict separation property allows us to
prove that every global weak solution converges to a single equilibrium as time
goes to infinity using the {\L}ojasiewicz-Simon approach. Finally, we study the
double obstacle limit for the problem with logarithmic potentials in the bulk
and on the boundary, showing that as the absolute temperature $\Theta$ tends to
zero, the corresponding weak solutions converge (for a suitable subsequence) to
a weak solution of the problem with a double obstacle potential.
