Most Emmys Won by a Single Show: 42
Unsurprisingly, SNL has won the most Emmys of any individual television show — it's been around for a long, long time, and each generation of voters seem to love handing Lorne Michaels' series statuettes. Its longevity and continued success, even as a variety show rather than a scripted series, is practically unheard of in the history of the medium. (If you discount soap operas, which are relegated to the separate Daytime Emmys.)
Turning to scripted shows, the comedy with the highest total number of Emmys is, unsurprisingly, the insanely-popular-in-its-day spin-off Frasier — a show remembered in large part for its intellectual sensibility, which made it catnip for sophisticated voters. Over the course of its run, the show won 37 Emmys with 108 nominations, including five Best Comedy victories; it outstripped its parent show Cheers, which took home 28 total Emmys with 107 nominations.
On the dramatic side, Hill Street Blues has 26 total Emmys with 98 nominations — edging out ER, which has 23 awards. But there will probably be a new king soon: Game of Thrones currently has 22 Emmys, with at least two seasons left to go in the show's easy-prestige-magnet run.