A simple horizontal trellis supported by four two-by-four boards is appropriate for a grapevine. Grapes need to hang to ripen and prevent rotting, and a horizontal trellis will provide support for the vine to prevent snapping. Reinforce your trellis with wire or use wood to help support the weight of the vine, as a full-grown grapevine can easily weigh 50 lbs.
Another appropriate climbing plant for a horizontal trellis is the cucumber. Cucumbers that ripen on the ground will often remain moist and rot before you have a chance to pick them. Growing your cucumbers on a horizontal trellis raised 2 or 3 feet off the ground is an easy way to allow the cucumbers to hang and ripen, and keep them off the ground away from insects and other pests.
If you are willing to raise your horizontal trellis 7 or more feet off the ground, you might want to consider constructing a pergola. A pergola is little more than a trellis, supported by vertical wooden posts or concrete pillars, and constructed out of wood or wire. Pergolas allow vines and climbing plants to grow freely along a trellis roof, which shades the area underneath and provides hanging fruit should you plant a grapevine.
You can use a horizontal trellis in conjunction with a dense-growing ivy such as English ivy to create a small shade garden. Train the ivy to grow across a horizontal trellis lifted off the ground 2 or 3 feet. Once the ivy fills out, you can plant shade-loving plants underneath, such as begonias or impatiens. The ivy will love the sun, and your more sensitive perennials will flourish in the partial shade.