There are several adhesives made especially for foam rubber sheets. 3M Company makes a spray on adhesive for foam called Fast7. Gorilla Glue is also used for gluing foams. Quick Stick is another product used for foam. Acoustic foam may be installed with any one of these glues. They glue the foam to wood or another foam surface.
Glue is used because it seals the foam directly to the surface of the wall or paneling. Nails or screws would puncture the foam and affect the soundproofing of the material. Foam is easily compressed as well. A nail or screw will not provide a secure installation without compacting the foam at the point of installation.
Gluing acoustic foam with an adhesive of any kind will provide a permanent solution. Before you glue the foam to a wall or your car interior make sure the foam is cut to the proper length and width. Also make sure you want to have foam in place for a considerable length of time. Removing the foam is difficult, and some adhesives leave residues behind.
When installing acoustic foam using adhesive, a spray-on glue is the easiest to use. This adhesive is simply applied to the surface of the foam, then the foam is pushed into place. Wait until the adhesive "sets up" for about a minute and becomes tacky before you press the foam into place. The foam will hold faster to the surface and the installation will be easier, especially on a ceiling or tough-to-reach surface.
Foam glue is sold at craft stores, hobby stores and some home-improvement stores. Gorilla Glue is available at nearly every hardware store. Specialty glues are available online by conducting a search for "foam glue." For gluing acoustic foam to metal or fiberglass, search marine stores for specialty epoxies. Glue guns are another alternative. Hot glue sticks may provide a better foundation for foam to be installed over drywall, fiberglass or plastic.