According to the Transportation Security Administration, hair straighteners are not among prohibited items, so you generally should be able to take them on a plane.
While hair straighteners are not prohibited on planes under federal law, individual airlines may have tighter rules. Contact your airline to see whether it will allow you to carry your straightener on the plane or whether you'll have to put it in your checked luggage.
Liquid Hair Products
Liquid hair products must follow the "3-1-1 rule" if you plan to carry them onto the plane: They must be in a container 85.1gr. or smaller; all of your containers must fit in one quart-sized clear plastic bag; and you may have only one such bag.
Hairspray and Other Aerosols
As long as they are nonflammable, you can pack aerosol hairsprays to help with hair straightening. If you plan to bring them on the plane, they must follow the 3-1-1 rule. If you plan to check them, place them in a zipped baggie so they don't spray inside your luggage.
If you are travelling internationally, you might not be able to use your hair straightener in your country of destination because of different outlet designs. You can, however, purchase outlet converters that will enable you to plug U.S. appliances into outlets overseas.
American outlets are 110-volt outlets, while those in Europe and other parts of the world are 220 volts. To ensure you don't blow a fuse or ruin your straightener, purchase an adaptor to use while abroad.