The renormalisation group theory of critical and tri-critical wetting transitions in three-dimensional systems with short-ranged forces, based on analysis of an effective Hamiltonian with an interfacial binding potential w(ℓ), predicts very strong non-universal critical singularities. These, however, have famously not been observed in extensive Monte Carlo simulations of the transitions in the simple cubic Ising model. Here, we show that previous treatments have missed an entropic, or low-temperature Casimir, contribution to the binding potential, arising from the many different microscopic configurations which correspond to a given interfacial one. We derive the full binding potential, including the Casimir correction term, starting from a microscopic Landau–Ginzburg–Wilson Hamiltonian, using a continuum transfer-matrix (path-integral) method. This is illustrated first in one dimension before generalising to arbitrary dimension. The Casimir contribution is qualitatively different for first-order, critical and tri-critical wetting transitions and substantially alters previous predictions for critical singularities bringing them much closer to the simulation results.