Employing the lattice-gas model with attractive interactions between nearest-neighbour particles, we have studied by Monte Carlo simulations the formation of islands in an adsorbed overlayer in two situations. The first situation corresponds to island growth when the adsorbed overlayer is quenched at t = 0 from high temperature in a disordered phase to a temperature below the critical one. Using a realistic algorithm of surface diffusion, and also the Metropolis dynamics for times up to 10 6 Monte Carlo steps, we show that in analogy with the spin-exchange kinetic Ising model the growth law for the average island size can be represented in this case at all the coverages as R( t) ⋍ A + Bt 13. The parameter A is found to be weakly dependent on coverage and on the details of the dynamics employed. On the other hand, the value and the coverage dependence of the parameter B are sensitive to the mechanism of diffusion. The second situation corresponds to phase separation in chemically reactive systems under steady-state conditions. As an example, we analyse the A+B→AB reaction, occurring via the Eley-Rideal mechanism, including A adsorption on vacant sites, A gas →A ads, and A consumption in collisions with gas-phase B particles, A ads+B gas→(AB) gas. For a realistic algorithm of surface diffusion, the average size of A islands is found to be rather low, even if surface diffusion is many orders of magnitude faster than the adsorption/reaction processes. In contrast, the Metropolis dynamics predicts the formation of relatively large islands or even mesoscopically-ordered structures (at θ A ⋍ 0.5). The physics behind these findings is discussed in detail.