Just a note: I'm in the process of keeping a tally of all Supreme Court cases from the 2012-13 term (the one that's coming to an end in the next few weeks) in which a law school clinic helped represent one of the parties to the case. I'm also going to start listing Supreme Court cases in which clinics contributed to amicus curiae (friend of the court) briefs.
All of these will be listed here, in a gradually expanding version of my original post on this subject. But from time to time I'll add new posts like this one, just to report that new cases have been added to the list.
So far, by the way, I know of 5 cases from this term in which clinics represented a party. Two of them, remarkably, are from a single clinic, the Stanford Law School Supreme Court Litigation Clinic, whose record for wins is 2 for 2! 5 cases may not seem like a huge number, but the Supreme Court, according to its FAQ page, "grants and hears oral arguments in about 75-80 cases" a year -- so 5 cases is actually about 6 % of all the cases the Supreme Court will have given its fullest consideration to during this entire term.
UPDATE: My count is now 6 cases, three of them from the Stanford Law School Supreme Court Litigation Clinic, whose record overall is 2 for 3. For the latest addition to the list, see the link above.