Final answer:
The method of execution not currently used in the United States is hanging, while lethal injection, the electric chair, and the firing squad are still utilized under scrutiny for adherence to the Eighth Amendment.
Step-by-step explanation:
The method that is not a contemporary form of execution within the United States is hanging. The Eighth Amendment prohibits cruel and unusual punishments, which includes various torturous methods used in the past. Contemporary methods like lethal injection, the electric chair, and even the firing squad are still utilized, albeit under scrutiny for their humaneness and constitutionality. Cases like Furman v. Georgia and Roper v. Simmons have defined the edges of what is acceptable under the Eighth Amendment, and while some states have abolished the death penalty, others continue to implement it in varying forms.