Judicial Watch has filed a lawsuit on behalf of State Department employee David Rodearmel claiming that Hillary Clinton isn't eligible to head that department (link):

The lawsuit maintains that Mrs. Clinton is constitutionally ineligible to serve as Secretary of State and that Mr. Rodearmel cannot be forced to serve under the former U.S. Senator, as it would violate the oath he took as a Foreign Service Officer in 1991 to "support and defend" and "bear true faith and allegiance" to the Constitution of the United States.

Under the "Emoluments" or "Ineligibility" clause of the U.S. Constitution, no member of Congress can be appointed to a civilian position within the U.S. government if the "emoluments" of the position, such as the salary or benefits paid to whoever occupies the office, increased during the term for which the Senator or Representative was elected.