A display may have an active area surrounded by a border area. The display may be a liquid crystal display having a liquid crystal layer sandwiched between a color filter layer and a thin-film transistor layer. The liquid crystal layer may be retained within the display using a ring of sealant that is dispensed along the border area on the thin-film transistor layer. The thin-film transistor layer may include at least a substrate, a dielectric layer formed over the substrate, a first planarization layer formed on the dielectric layer, and a second planarization layer formed on the first planarization layer. A first continuous trench structure may be formed along the border of the display to help prevent moisture seepage. A second trench structure that is separate from the first trench structure may be formed along the border of the display to help provide proper sealant adhesion.