I realize there are a lot of these, so I've kind of arbitrarily picked. Feel free to share your favorites (or least favorites, I suppose) in the comments!
1. The Tortellis, spun off from Cheers. Based on Carla's ex-husband Nick and his new wife and family, it was only on the air for three episodes in 1987.
2. Day by Day, spun off from Family Ties. Loosely spun off, anyway. It starred Thora Birch, Julia Louis-Dreyfus and Courtney Thorne-Smith, among others. The connection to Family Ties is that the dad in Day by Day apparently was a college roommate of Steven Keaton.
3. Just the Ten of Us, spun off from Growing Pains. I remember Just the Ten of Us and never realized the connection to the Seaver family. The Lubbock family patriarch, Graham, was the gym teacher at Mike and Carol Seaver's high school. He ends up losing his job at that high school, though, and gets another job at an all-boys Catholic school in California. It lasted three seasons, so compared to others on this list, it wasn't such a disaster.
4. Blansky's Beauties, spun off from Happy Days.
In 1977, it seemed like a good idea to make a sitcom based on Howard Cunningham's cousin from Las Vegas. The cousin, Nancy Blansky, was in charge of a bunch of Las Vegas showgirls living together in an apartment like it was a dormitory. Only 13 episodes were made.
5. Grady, spun off from Sanford and Son. Based on Fred Sanford's widower pal Grady (go figure), this one was cancelled after 12 episodes. Great, now I'm going to have the Sanford and Son theme song stuck in my head for the rest of the day.
6. Sons of Thunder proves that even being associated with Chuck Norris doesn't guarantee series gold. Spun off from Walker, Texas Ranger, Sons only lasted six episodes. After Walker's buddy, Reverend Thunder Malloy, dies, his son comes home.
7. Mr. T and Tina, spun off from Welcome Back, Kotter. No, not the Mr. T. Mr. T. was Taro Takahashi, a Japanese inventor played by Pat Morita. He lives with a free-spirited American girl names Tina and hilarity ensues. At least, it did for five episodes in 1976. Pat Morita was also in Blansky's Beauties, oddly enough. Good thing he eventually found the Karate Kid!
8. Living Dolls, spun off of Who's the Boss. You may know Leah Remini from King of Queens, and I may know her as Stacey Carosi from Saved by the Bell, but before both of those she starred in this brief sitcom. Leah played Charlie Briscoe, one of Samantha Micelli's old friends. She is discovered by a modeling agency and the rest is sitcom history. Except not: it was canceled in 1989 after 12 episodes. Interesting trivia, though "“ Vivica A. Fox and Halle Berry played the same character on the show (kind of like the mulitple Beckys on Roseanne).
9. Checking In, spun off of The Jeffersons, which was spun off of All in the Family. Pretty much every sitcom spun off from All in the Family did wonderfully on the small screen "“ Maude, Good Times, The Jeffersons. But not Checking In. In it, the Jeffersons' maid got a new job as the head housekeeper at a swanky hotel. She checked out after four episodes in 1981 (sorry, couldn't help the stupid pun).
10. Fish, spun off of Barney Miller. Abe Vigoda in his own sitcom? How could this fail? Phil Fish was a detective on Barney Miller, but his own series focused more on his life with his wife and five adopted kids (one of which was played by Todd Bridges). It lasted two seasons before fizzling out.