There were three spin-offs from the Mary Tyler Moore Show: Rhoda; Phyllis and Lou Grant (two comedies, one drama).
Is that a record for number of spin-offs from one show?
All in the Family had five:
The Law and Order franchise has it beat. After the original show came:
[ol]
[li]L&O: Special Victims Unit[/li][li]L&O: Criminal Intent[/li][li]L&O: Trial By Jury[/li][li]L&O: UK[/li][li]L&O: Los Angeles[/li][/ol]
nm
LOVE, AMERICAN STYLE begat HAPPY DAYS, which in turn begat MORK AND MINDY and LAVERNE AND SHIRLEY and JOANIE LOVES CHACHI and BLANSKY’S BEAUTIES.
Six, you forgot Good Times.
In the wake of the original STAR TREK and STAR TREK: THE ANIMATED SERIES, we got STAR TREK: THE NEXT GENERATION as well as STAR TREK: DEEP SPACE NINE as well as STAR TREK: VOYAGER as well as STAR TREK: ENTERPRISE, with the latest spinoff apparently on its way in a matter of months.
Do reality shows count? I’m always hearing my sisters talk about Real Housewives so I just went and counted them. Not counting the original show (Orange County), there’s:
8 Cities
14 shows “spun off” from other RH shows (ie Vanderpump Rules, Bethenny Ever After)
9 International RH shows.
That would be 31, if we’re counting reality shows.
Some that match or beat it, just off the top of my head…
All in the Family had 4 - The Jeffersons, Gloria, Maude and 704 Hauser (arguably Archie Bunker’s Place, too). The Jeffersons and Maude both had spinoffs of their own (… Both about their maids, actually) - Checking In (where Florence went to work at a hotel) and Good Times (Maude’s maid Florida’s home life, after she moves away).
MASH technically had 3, although WALTER never got more than the pilot made (but it did air, so it still counts)…The other two (AfterMASH an Trapper John, MD (also a drama) did well though (2 seasons and 6 seasons, respectively).
Star Trek has The Animated Series, The Next Generation, Deep Space Nine, Voyager, Enterprise, and the upcoming Discovery, though the distinction between spinoff and sequel/prequel is hard to define for them. But if you count them as spinoffs, that’s 6.
Happy Days had Laverne & Shirley, Blansky’s Beauties, Mork & Mindy, and Joanie Loves Chachi, so 4, even without considering the Fonzie cartoon.
[Edit - Jesus, I spent too long writing this post and looking up titles of ones I’d forgotten - 704 Hauser, Checking In, WALTER, and Blansky’s Beauties - and got beaten to most of these.
Good Times is a spinoff of Maude, so interesting, but not an AitF spin-off, per se, and you also forgot Checking In. 
I don’t really feel like the Law and Order franchise are “spin offs”. More of a repeated formula. But I don’t make the rules. 
I was thinking Happy Days too but a couple of them seem to be “backdoor pilots” which I have a problem with calling a real spinoff. Which brings me to a thought: Can any show top the backdoor pilot chain of JAG -> NCIS -> NCIS:New Orleans&LA?
We just did this a few months ago. Can’t find the thread though.
Do international adaptations count as spin-offs? For instance, “All in the Family” was an Americanized version of a British sitcom “Man About the House”.
If such adaptations count, then “the Office” would be a strong contender with eight official versions.
There are also a host of international versions of “Charlie’s Angels”, though I don’t know how many.
There is, of course, a Wikipedia article about this:
You make up your own mind about what really counts as a spin-off. The article lists the most spin-offs from Popstars. These shows were basically copies for other countries.
I’d consider a spin-off to be a new show including cast member(s) from the previous show. Hence the ‘spin-off’; a character spins off to create a new show in the same ‘world’.
I’m not familiar with Real Housewives or L&O enough to say, but if they’re all new casts in a new location than I wouldn’t call it a spin-off. Of course, by my definition, even Star Trek: TNO isn’t a spin-off so much as a continuation on the formula. I’m okay with that but I could see where others wouldn’t be.
In that case Love American Style cannot count in the Happy Days bloodline
I think you got your wires a little crossed, there. All in the Family was a remake of the British Till Death Us Do Part. Man About the House was remade in America as Three’s Company.
Should Blanksy’s Beauties and Out of the Blue be counted as Happy Days spinoffs? The only reason those shows’ main characters appeared on HD was to promote them for their own shows. It’s like claiming that Hello Larry was a spinoff of Diff’rent Strokes.
Does the common character need to be the main character of the new show? Because if not, then we can at least count Deep Space Nine as a spinoff of Star Trek: The Next Generation, with O’Brien (and later Worf) in common.