To remove chewing gum from carpet, you can use your hair dryer. Just heat the gum, being careful not to melt any carpet fibers. Then you can use a piece of plastic wrap or a plastic bag to lift the softened gum away. Allow it to stick to the plastic, and pull it up. This will often get most of the gum. For oily or greasy stains, use dry cleaning solvents or in exceptional cases, methylated spirit.