They seem to sprout up everywhere, sometime going unnoticed until they have covered an apple tree. We cut them down, but they just keep growing. Is there any beneficial use for them? Birds probably like the grapes. I sometimes think I will keep one growing and somehow make a giant wreath or basket or something else sculptural out of it, but so far, it always wants to do its own thing!

You can eat the grapes, make jelly out of them, make wine out of them but unless you kill the root they are not going away.

If your apple tree is big enough, you could use it as a living trellis for the grape vine. Vines are one of the seven layers in natural forests, and they want to be around your trees. You will constantly be fighting them if you don't let them grow. Also, grapes only grow on first year growth, so if you cut the vine back after it goes dormant, you should be able to increase the yield.

Around here the squirrels strip the bark and use it for nests

I use it as a bait for the Japanese beetles, they love it and leave other garden plants alone.