The "4291" puppy in One Foot In The Grave.
The crab that crawls up his neighbours shorts also.
All the aliens species in Red Dwarf, especially the polymorph who turns into a rabbit, a snake and a chicken at one point.
If dead animals count then all the turkeys and chickens eaten at Christmas Specials and the like.
Father Jack's pet brick.
The pig from Blandings.
All the farm animals from The Green Green Grass.
The pig from Max And Paddy's Road To Nowhere and the cow they run over.
Jerry The Gerbil which Damien Had In Only Fools & Horses
Quote: lofthouse @ 7th April 2016, 8:01 PM BSTMongy the dog from Bread
Ah good old Mongy. He was mentioned a lot but rarely on camera. It's been a long time since I watched that so might have to get the box set.
A few more...
The pig Mel Smith holds in Bambi (the young ones)
The elephant Tony Robinson thinks is Joseph Merrick (the young ones) "I am an elephant, you know" 
Colin the Dachshund in Dish & Dishonesty (blackadder III)
Defor the Dog in Second Thoughts.Quite amusing sit com but went off a bit when Geoffrey Whitehead left.
Onslow's dog in Keeping Up Appearances, the one Hyacinth used to fall into the bush from.
The Ram Mascot in Dad's Army
Mummy the dog, whom Jeremy kills and eats in Peep Show
David Brent's dog, Nelson
Nana's budgie, Robson in Royle Family
The Captain in Phoenix nights had a dog
I think Max and Paddy had a pet pig or something like that
And Victor Meldrew thought he was getting a pet dog called nippy, but it was a stuffed dog.
Harry Grout's budgie - Porridge
The horses in To the Manor Born
Did Flo in Citizen Smith have a budgie?
Ferrets - Last of the Summer Wine
Goldfish - George and Mildred
Mildred's dog Truffles - George and Mildred
Vienna the cat - Rising Damp
Chickens and bees - The Good Life
Dorothy's dog - Two's Company
Kitten Kong - The Goodies
Harold Steptoe's Goldfish, Charlie
Super-Hans snake in Peep Show
The Neds dug that jumps off the roof in Still Game
Denzil and Corrine's canary in Only fools.
Bertie in To The Manor Born
Miranda episode 'Dog' had 3 dogs, plus there was another episode with a flashback and a dog called 'Jaspar'
Mr Barker was the head teacher in the first series of Class Dismissed (not a sitcom I know, but it was very funny!)
Owen had a farm and often mentioned his cows and sheep in Vicar of Dibley
Whilst Eddie in Frasier has already been mentioned, lets not forget that (I think unnamed) Cockatoo which Nialls owned and seemed a little scared of, bless him.
Tons of animals in friends. Joey's Chicken and Duck, Marcel the Monkey.. and lots of other cameos, such as that Spyhnx cat Racheal buys in one episode, accompanied by Ross's immortal line of 'Why is it inside out?'
Of course the Good Life had its fair share too - there's that brilliant episode where the rooster catches the bus and Tom goes after it.
The horses in the Gunmen of the Apocalypse episode of Red Dwarf. I wonder if they were a bunch of cats taped together.
Fred had lots of horses in Hi-De-Hi.
The chickens that the Dad's Army squadron attempt to catch.
From the film, the horse that ends up going down river.