We may already have covered this, but I'm wondering about shows that grew out of other shows? The most obvious example is Cheers giving birth to Frasier. Other examples include Swiss Toni and Grass coming from The Fast Show, and Green Green Grass coming from Only Fools and Horses...
Any other examples? Either American or British?
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Man About the House - Robin' Nest, George and Mildred
On the Buses - Don't Drink the Water
Please Sir - Fenn St Gang, Bowler
Second Thoughts (BBC radio) -> Second Thoughts (ITV) -> Faith in the Future
The Growing Pains of PC Penrose -> Rosie
Nearest and Dearest -> Not On Your Nellie (although indirectly)
Dad's Army -> It Sticks Out Half a Mile
Coronation Street - Pardon the Expression
I was going to say Yes Minister and Yes, Prime Minister, but the latter was more of a sequel than an 'offshoot'.
Mork and Mindy?
Coronation Street - The Brothers MacGregor
Desmond's -> Porkpie
Happy Ever After - Terry and June
Same basis that Aaron used to match Nearest & Dearest with Not on Your Nellie 
Lou Grant and Rhoda came out of the Mary Tyler Moore Show.
Quote: Alan C @ June 12 2008, 2:17 PM BSTHappy Ever After - Terry and June
Same basis that Aaron used to match Nearest & Dearest with Not on Your Nellie
I was going to comment on that, but then you edited. So I'll add:
Sykes and A... -> Sykes
Oh Brother - Oh Father
Till Death Us Do Part - Till Death - In Sickness and In Health
All Gas and Gaiters -> Oh, Brother! -> Oh, Father!
Quote: Aaron @ June 12 2008, 2:21 PM BSTAll Gas and Gaiters -> Oh, Brother! -> Oh, Father!
Not letting you get away with that one - All Gas has nothing to do with the other two except Derek Nimmo was in them
Romany Jones - Yus My Dear
Are You Being Served? -> Grace and Favour
The Fall and Rise of Reginald Perrin -> The Legacy of Reginald Perrin
Agony -> Agony Again
(Although more sequels than offshoots really.)
And I suppose one could even argue Hancock's Half Hour -> Hancock, and indeed Hancock's Half Hour -> Citizen James.