The wetting state of a simple liquid on a solid substrate, as summarized by Young's equation, is dictated by the interfacial energies of the different phases that coexist in the system. For simple fluids, rotational symmetry gives rise to symmetric droplets around the axis perpendicular to the substrate. This is not the case for nanostructured fluids, such as block copolymers, where the inherent thermodynamic ordering forces compete with surface tension. This competition is particularly important in nanoscale droplets, where the size of the droplets is a small multiple of the natural periodicity of the block copolymer in the bulk. In the nanoscale regime, droplet shape and internal structure arise from a subtle interplay between interfacial and bulk contributions to the free energy. In this work, we examine the consequences of surface polymer interaction energies on droplet morphology through a concerted simulation and experimental effort. When the block copolymer is deposited on a neutral substrate, we find noncircular arrangements with perpendicular domains. However, when a preferential substrate is used, the resulting morphology depends on droplet size. In large droplets, we observe bottle-cap-shaped structures with a ring of perpendicular domains along the perimeter, while small droplets exhibit stripes of perpendicular domains.