Electronic devices are provided that contain wireless communications circuitry. The wireless communications circuitry may include radio-frequency transceiver circuitry and antenna structures. The antenna structures may include antennas such as inverted-F antennas that contain antenna resonating elements and antenna ground elements. Antenna resonating elements may be formed from patterned conductive traces on substrates such as flex circuit substrates. Antenna ground elements may be formed from conductive device structures such as metal housing walls. Support and biasing structures such as dielectric support members and layer of foam may be used to support and bias antenna resonating elements against planar device structures. The planar device structures against which the antenna resonating elements are biased may be planar dielectric members such as transparent layers of display cover glass or other planar structures. Adhesive may be interposed between the planar structures and the antenna resonating elements.