We grow winter squashes (buttercup and delicada varieties) on our roof of our house (which is underground and the roof is organic material)...thus we have a 'living roof'. The squash leaves stand a foot to a foot and a half above the roof shading it from the summer sun and aiding in the cooling of the house.

I grow hop vines to shade and cool my house . Theyare not annuals but they do die back to the ground completely in the winter .I train them on strings instead of a trellis so i can cut the whole thing down for the winter to let the sun in.I have four 3'x6' windows on my south wall so shade and cooling are very important to me, so is the solar heat in the cold months. the vines take a couple years to get established but they have no trouble reaching 15 feet even in my short season. For a by-product they produce the cone like flowers (hops) used in brewing beer .

I would never ever grow a vine on the roof. Simply asking for trouble. An asphalt shigle roof would mildew from the retained moister and the shingles could be seperated one from another by searching tendrils. Metal roofing would be likely to rust.
Also with nothing for the vines to 'hold' to they would be sweep off the roof by a good wind. More trouble with all that weight pulling the vine away from the trellis.
Now with that said I have seen sheds and small structures for animals, it was a chicken shack, with a trellis attached to the roof and vines growing up on it. Helped keep the place cool you bet but that roof will not be long in needing repairs.
Stick with shading the south and west side of your house. It will make a big difference in keeping you house cool. Plant a deciduous tree or two on the south or s/w side of the house. You may even wish to plant two different types of trees in those spots, a fast grower for fast shading (but will more than likely be a short lived tree)and a slower growing tree for long term shading (will be with you for years and years but will need time to grow to any real height).
My favorite vine is maypop. I harvest the fruits and make a most wonderful drink from them. Another fast growing vine is LabLab (hycanith bean) which is a beautiful vine and yes the beans are edible. Kiwi is a fast fast fast to grow but you had better have a VERY strong support and be prepared to do year maintance on this vine. Others ... grapes (muscadines down here) and malabar spinach (you would need to resow yearly that far north I'm guessing) has a wonderful mild flavored edible leaf and can handle the hot/humid summers of the South.
Dems my picks.
Mike

Many of my neighbors grow Virginia Creeper as living awnings and on the sides of houses. They are perennials and would have to be controlled, but as vines go, they are pretty well behaved. (Saying something is a well behaved vine is damning with faint praise though, so be prepared to control its agressive tendencies, or leave it out). Not edible, but native, not invaisve or poisonous and an important wildlife food source. They can be confused with poison ivy since the young leaves look similar, but mature virginia creeper leaves have FIVE leaflets as opposed to PI, which has three. Also, PI vines have fine hairs on them and Virginia Creeper vines are much less hairy, more tendril-like since the vine is related to the wild grape. If you find one in the wild you could transplant it for free, they grow all around waste places, but be sure it's not poison ivy before you start digging!

Here is a link that might be useful: U of Arkansas Cooperative Extension Virginia Creeper Description.

See the website mentioned in our profile for a photo example of our 'living roof'....I posted it below, as well.

Here is a link that might be useful: living roof and more

I am going to try Scarlet Runner Beans in planters, then trained up strings to the bottom edge of my patio cover. The cover is a pole and tarp type. If they get taller than 8 feet, they still have to "crawl" another 10 feet over the tarp before they reach my gutters/shingles. This porch faces west and is concrete so is very hot in the summer. Plan to "harvest" my shade about the time the weather is cooling down.

Moringing Glory is an EXTREMELY INVASIVE weed here in Portland Oregon. Please know your enemy people, and use native plants whenever possible.

"Moringing Glory is an EXTREMELY INVASIVE weed here in Portland Oregon. Please know your enemy people, and use native plants whenever possible."

I agree that you really need to be mistrustful of your area's invasives (personally, I won't grow any invasive that takes over via seed, and anything that takes over via runner goes in my container garden). The good news is, some varieties of MG (like "Heavenly Blue") don't really have volunteers from seeds, so they really are just a one-time annual that must be re-planted every year. : )

Here's a photo of my Runner beans. Not much shade but they sure look pretty!!
Next time, I would use more strings, closer together. These have several climbing up each string, but they are too far apart to provide the shade I was looking for. Also, start them earlier!! It doesn't look like it in this pic , but they are 10' from the house. There's a tarp style cover over the patio.
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if you want to grow vines on your roof and are afraid of moisture damage, simply lay fown a waterproof tarp first, white ones wood be best to help keep heat down inside the house and also reflect light back to the plants.and then add some simple string for support wires. just stay away from things that use roots for its support when growing like ivy.

im starting to grow grapes and kiwi on my roof.

My very late 2 cents worth:

As a former long-term resident of Georgia, please, please, PLEASE don't plant kudzu. It is one of the most disastrous examples of an introduced species taking over entire ecosystems and choking everything in its path. Kudzu will bring down trees, powerlines, and will completely f* up your roof.

Why not use a seedless grape? We have one, "Himrod", which has provided tons of wonderful sweeet grapes for years. Put it on a trellis made of steel pipe (recycling, natch)about 18" out from your house/roof and you will be rewarded with shade in summer, grapes in fall and a beautiful shaggy vine in winter.

For providing shade on my porch and around my outdoor kitchen, I use hops, angel darts and morning glories on heavy hemp twine net suspended from supports that aren't attached to my house.

I agree with Mike, aka bigeasyjock, about not growing vines ON your roof, IF you have a traditionally built, wood-frame house, with asphalt shingles. Roofs were not designed to support any additional weight. Think about it. It's the very top of the building, nothing else is going on it, why engineer it to support any more weight than it has to?

Also, using a waterproof tarp is also a bad idea. Moisture will condense on the underside of that tarp and mildew and other nastiness will begin growing there. Now grow some heavy vines on top, vines that perspire like all green leafy plants do, and you've invited lots and lots of insects to make your roof a nice comfortable home.

Seriously, a nice deciduous tree providing shade for the south and west sides of your house will help a lot more than vines on the roof.

Vines are wonderful, but they are high maintance to grow responsibly, and are best left on vertical supports around the home, not on the home itself.

Oh, and before planting anything. ANYTHING, check with your state Ag dept and make sure it's not listed as a noxious weed. I'm pretty sure Kudzu is all over the south.

you could always seal the tarp to the top of the roof with waterproof tacking. some folks dont have room to plant trees. If your roof cant hold an added 30 to 40 lbs of weight you have got some serious structural problems that need emmediate addressing.