Block copolymer brushes where chains are grafted onto the substrate have been of special interest due to their capability to form various self-assembled nanoscale structures. The self-assembled structures depend on complex system variables, including block copolymer composition, segregation strength, grafting density, selectivity of top and bottom surfaces, and solvent quality. In spite of extensive previous efforts on understanding and controlling the microphase separation of diblock copolymer brushes, an individual study focuses only on the subset of the parameter space of system variables. This work systematically explores the full parameter space within a single simulation framework of a coarse-grained model with a generalized Hamiltonian. The topologically unconstrained free surface allowed in the model enables us to investigate brush systems under versatile and more realistic conditions. We show that melt brushes with non-selective surfaces can form previously unexplored structures such as “void” and “curvy” phases at a moderate grafting density; special emphasis is placed on the system's evolution to make such structures. Our simulations demonstrate that the surface selectivity further enriches the phase behavior with a diverse range of phases obtained even at a single composition. We also find out that the phases with a similar morphology from the top view can vary significantly in their internal structures; such variations are discussed when examining the effects of grafting density and surface selectivity. Finally, we studied the influence of exposure to a non-selective solvent on the morphology of block copolymer brushes and compared the topographical variation on the top at varied solvent qualities.