We employ an order parameter approach to investigate melting in clusters bound by the Lennard-Jones potential containing 13, 55, and 147 atoms. We find well-defined Landau free energy barriers between solidlike and liquidlike states for the two larger clusters. A barrier is also revealed in an approximate analytical calculation using only information derived from the potential energy surface. For the two smaller clusters the order parameters are calculated for a large number of local minima. This helps us to interpret the Landau free energy calculations and to comment upon the suitability of the various order parameters for the cluster melting process. Systematic quenching offers us further insight into melting events for the 55-atom cluster. Finally, we elaborate further upon the relationships between S-bends and probability distributions in different ensembles.