I've just added one more case to the list I've been compiling of Supreme Court decisions from last term (2012-13) in which law school clinics helped represent a party -- now a total of 8 out of 78 cases decided on the merits by the Court that year. Arguably these numbers understate the clinics' influence, since they don't include amicus briefs that clinics may also have filed (though there are lots of those briefs nowadays). I don't know whether the 2012-13 numbers reflect an ongoing phenomenon, but I hope to keep an eye on this year's cases as well and try to find out.
The presence of law clinics in the Supreme Court may startle both clinical and nonclinical faculty: clinicians because most clinics focus largely on interpersonal skills of lawyering, while the Supreme Court litigation experiences presumably are mainly about writing, and nonclinicians because writing a Supreme Court brief is the sort of elite, intellectual practice work that even those who most strongly emphasize the role of doctrine and scholarship in law school would likely validate. That makes the role of clinics in the Supreme Court important pedagogically as well as practically.
Meanwhile, congratulations to the clinical teachers and students who did this work!