During the 1980s and 90s, Selberg occasionally lectured on what he called "the hybrid trace formula". a special kind of trace formula associated with Hilbert modular-type groups. In 1984, for instance, he lectured on this topic at both Stanford and the AMS-MAA-SIAM Joint Summer Conference on Trace Formulas held at Bowdoin College. Aside from the final entry below, the attached scans are basically copies of his transparencies together with some fragments of his working notes from these 2 lectures.
Additional places where he lectured on this matter include Stillwater(1984), Tata Institute (January 1988), and Institut Mittag-Leffler (1995). His talk at the Tata Institute was given during the Ramanujan Centenary; file 4 appears to be its abstract. The results themselves date back to the early 1970s or before.