Football - Easily Tecmo Super Bowl. It improved in nearly every aspect from the original and had a unique feel and presentation that was unmatched for a long time.
Hockey - Blades of Steel wins here. It had fights, mini-games, and a easy to understand control system.
Baseball - Baseball Stars is the champion (followed by Little League Baseball). It allow customizable teams and players and even the ability to improve your players based on your performance. Absolutely great game that wasn't matched until the World Series Baseball titles on Genesis.
Basketball - Arch Rivals. I know, some of you are saying Double Dribble, but Double Dribble lacked a lot of appeal that the games above brought to the table. Arch Rivals was innovative and new, basically being the precursor to titles like NBA Jam later.
Other - Pigskin 621 A.D. For those of you who visited the arcade when you were younger, you saw this game and were amazed. It was a mix of football and rugby, set in medieval times that included weapons, obstacles, players, and yes, even trolls. It was fun and innovative, but was definitely a quarter monster. Expect to lose money fast on this one, but it was worth it.