The National Law Journal reports on law school Supreme Court clinics. Stanford Law School is credited as being the first to start a Supreme Court Clinic:

The first Supreme Court clinic was formed at Stanford Law School in 2004. It has worked on 63 cases, including 20 merits cases. [Related article: "Stanford's Supreme Court Clinic Setting a Record."] The Supreme Court recently accepted two cases from the clinic, one involving mandatory minimum drug sentencing and the other centered on a voting rights issue in Alabama. Burgess v. U.S., No. 06-11429, and Riley v. Kennedy, No. 07-77.