Final answer:
The U.S. Supreme Court case that declared habeas corpus a fundamental right and struck down parts of the Military Commissions Act was Hamdan v. Rumsfeld. The Court's decision in 2006 and subsequent ruling in 2008 emphasized the importance of habeas corpus for detainees, including those designated as enemy combatants.
Step-by-step explanation:
The Supreme Court Ruling on Habeas Corpus Rights
The case where the U.S. Supreme Court struck down parts of the Military Commissions Act and declared habeas corpus to be a fundamental right was Hamdan v. Rumsfeld, in 2006. This landmark decision stated that the setup of military tribunals for detainees at Guantanamo Bay violated U.S. federal law and the Geneva Conventions. Additionally, in 2008, the Court further affirmed the rights of detainees by ruling parts of the Military Commissions Act unconstitutional, effectively recognizing the right of enemy combatants to invoke habeas corpus protections to challenge their detention.
The writ of habeas corpus ensures that the government must provide a legal cause for detaining an individual. Its suspension has been a matter of debate, particularly concerning individuals detained as part of the War on Terrorism, such as those held at Guantanamo Bay. The Supreme Court has consistently upheld habeas corpus as a fundamental right, integral to the rule of law and the protection of individual liberty.