After doing some research, I determined that the grapes growing on my shrubs were fox grapes (wild grapes), common in the US. They are pretty sour to eat off the vine, but I decided to try and make some jelly with them. I am really glad I did, as the jelly is delicious!
So I have decided that these vines are not useless after all. Late this fall I will trim them back and then give the shrubs a real hard trimming. Then next year when the vines grow across the top of the shrubs I will be able to reach them. As they are now, quite a lot of it is actually hanging down into the neighbors yard like a curtain. Maybe when I see them I will ask if they have grapes over there and if they want them or not. My bet is they don’t really care about them at all and are just to lazy to cut them. I will also give them a jar of jelly!