This Monte Carlo study on a simple cubic lattice investigates ternary mixtures of triblock copolymer-solvent-solubilizate systems consisting of solvent molecules, singly dispersed copolymer molecules, singly dispersed solubilizate molecules, and micelles of all sizes containing various amounts of insoluble blocks and solubilizates. In analogy with low molecular weight surfactant systems, two model structures are considered, one for dissolution and the other for microemulsion. Various approaches are carried out to identify and differentiate these two models. Under weak solubilization conditions, solubilizates are uniformly dissolved in the micellar cores. A gradual transition to the second model, where solubilizates accumulate in the centers of micellar cores with surfaces decorated by copolymers, occurs as solubilization becomes strong, due to either the presence of large quantities of solubilizates or the high incompatibility between immiscible species. As a consequence the solubilization process takes place in two steps, with each step arising from a distinct mechanism. The Langmuir adsorption isotherm is a good approximation for weak solubilization, while significant deviation from this isotherm is seen at strong solubilization.