A roof with curling shingles is not pretty in any sense of the word. It is also a potential major problem, since shingles that do not lay flat allow for rainwater and melted snow to get underneath the shingles and begin damaging the wooden structures below. While you could replace the entire roof by hiring a roofing contractor, there is something else you can do: address the issue with the curling shingles. There are four ways to get these shingles to stop peeling back and secure them.
Nail Them Down
The most obvious patch solution is to flatten the troublesome shingles and then nail them down. This is fairly effective. They may begin to curl up again if the nails do not hit a stud board underneath. (In this case, the "studs" would be the trusses, which are evenly placed to create effective support for your roof.)
If you can, attempt to find the trusses via a stud finder. If the trusses are near or right under the curling shingles, then nail down your shingles right in those spots. If not, then you may want to try a different solution.
Use Asphalt Emulsion as "Glue"
You are trying to fix curling asphalt shingles, and you want to prevent water from getting underneath. As such, you can use an asphalt emulsion as a sort of "glue" to flatten the shingles. Using a thick, wide brush, paint the emulsion on under the curling shingles. Place a standard red brick over the top to weigh the area down until the emulsion dries. Then remove the bricks so that they do not fall off the roof and clobber you or someone else in the head.
There are waterproof roof caulks you can use, too. Use them to fill the spaces under the curling shingles, then use a rubber mallet to hammer it flat. The flattened caulk dries fast and acts as a glue to hold the shingles down and in place.
Cut the Curling Shingle Loose and Replace It
When the shingles on your roof are applied in one foot by three feet sheets, you can cut the offending shingle loose, cut a replacement shingle from a fresh sheet, and insert that new shingle in place of the old one. Of course, you could remove and replace the entire sheet too, but that comes with a little extra work. A single slab pack of roofing shingles and a utility knife is a lot cheaper than a new roof.