Your fence doesn't have to be just a fence. You don't even have to settle for landscaping that is merely attractive (although that is a perfectly acceptable choice). A great way to make use of the space around your fence is to make it into a mini garden. Rather than having to maintain rows of plants and giving up a corner of your yard, this gives you easy access to the plants and allows you to keep them in a space that isn't otherwise usable.
Veggies That Grow on Vines
Cucumbers, tomatoes and peas all need a vertical surface to grow on. When you grow them in the garden, you have to tie them up on stakes or poles in order to keep them happy. This property makes them the best candidates for growing along a fence line. You can have them grow up the fence instead of using poles or stakes to hold them up.
How well this will work depends on the type of fencing you use. Chain link fencing may be the perfect material for this project. The metal will resist corrosion from the plants, and the links are the perfect place for plants to grab onto. If you have a stone or wood fence you want to keep pristine, you can still use this technique to grow some veggies for your table. Just install trellises a few inches in front of the fence. The plants will attach to the trellises instead of the fence. A few tendrils may make it over to the fence surface, but you can easily remove them.
Strawberries Are Easy to Grow on Vertical Surfaces
Strawberry plants do best when there is plenty of space between the plants, but this can make them inefficient to grow on a flat surface. The solution has been to breed varieties that can be planted just about anywhere, using small pockets of soil for each plant and collecting them in a tower or hanging basket. While this is an effective solution, a hanging basket still isn't going to contain enough strawberry plants to grow more than a bite or two at a time.
Your fence is a great place to put in a large number of plants using this technique. You will need to build a structure to hold these plants, but there are plenty of directions online that will give you ideas on how to build the structure at minimal cost. From there you just need to attach it to your fence and plant the strawberries. Within a couple years you will be reaping the rewards in the form of delicious homegrown strawberries that are far better than anything you can find in your local grocery store.
Apples are another fruit that is easily grown at home. The problem with apple trees is not the amount of space they take up, but that harvesting the apples is difficult. Any that aren't picked simply fall to the ground, where they rot or attract critters you'd rather not have in your yard. An interesting technique, called espalier, was developed in Europe to counteract these issues.
If you have ever seen pictures of vineyards with vines trained along wires to maximize production within a small space, then you can easily understand how this works, as this is essentially the same technique practiced with apple trees. As the tree grows, the branches are tied to a fence, wire or trellis, and any branches that don't fit the pattern are trimmed. Over the course of few years, you get a healthy, fruiting tree that is completely flat.
The possibilities are endless, but these are a good place to get started. Once you've established a good growing pattern in the area, you can add some more challenging plants that are favorites at your dinner table.