Bicomponent block copolymers are known to exhibit rich phase behavior in systems containing one or two block-selective homopolymers or solvents. In this study, we combine these efforts by investigating ternary blends composed of an ABA triblock copolymer, an A-selective homopolymer and a B-selective oil. A styrenic thermoplastic elastomer is selected here because of its ability to form a physical network upon microphase separation and thus impart significant elasticity and toughness to such blends. Synchrotron small-angle X-ray scattering is employed to classify the nanostructures of blends varying in composition, homopolymer molecular weight, and oil type, and the results are used to construct ternary morphology diagrams that reveal the phases present at the glass transition temperature of the styrenic endblocks. Of all the classical and complex morphologies commonly observed in binary copolymer blends and solutions, only the bicontinuous gyroid consisting of styrenic channels in a mixed midblock/oil matrix is consistently absent. Variations in nanostructural dimensions with blend composition are provided for selected morphologies.