  We present a study of time-independent solutions of the two-dimensional
discrete Allen-Cahn equation with cubic and quintic nonlinearity. Three
different types of lattices are considered, i.e., square, honeycomb, and
triangular lattices. The equation admits uniform and localised states. We can
obtain localised solutions by combining two different states of uniform
solutions, which can develop a snaking structure in the bifurcation diagrams.
We find that the complexity and width of the snaking diagrams depend on the
number of "patch interfaces" admitted by the lattice systems. We introduce an
active-cell approximation to analyse the saddle-node bifurcation and
stabilities of the corresponding solutions along the snaking curves. Numerical
simulations show that the active-cell approximation gives good agreement for
all of the lattice types when the coupling is weak. We also consider planar
fronts that support our hypothesis on the relation between the complexity of a
bifurcation diagram and the number of interface of its corresponding solutions.
