haffyfan and some of the VIPs have been collating a list of all the current companions and the bonus each one gives. It's so useful when choosing which to give to your horse so thank you to them for allowing it to be shared with everyone on the game.
Dressage
Hen +1
Squirrel +2
Goat +2
Armadillo +2
Armadilla +2
Weta +3
Giraffe +3
Butterfly +3
Rabbit +3
Hummingbird +4
Eared seal Piñata +4
Swallow +5
Monkey +6
Delfín +6
Orca +6
Camel +6
Kangaroo +8
Sheep +8
Cat +9
Bear +9
Ladybug +9
Zebra +10
Penguin +10
Turtle +10
Llama +10
Lil' Octopus +11
Frog +12
Phoenix +12
Behemoth + 13
Owl +13
Griffin +14
Dog +15
Bee +15
Green, Red, Blue and Purple Dragon +16
Ghosts +18
Jumping
Kaeru +2
Cardenala +2
Cardenal +2
Ōkami +3
Clownfish +3
Mantarraya +4
Mantarrayo +4
Kaeru no Senshi +4
Blue Whale Piñata +5
Okuri Inu +6
Gouldian finch +6
Orca Piñata +8
Namazu +10
Fox + 10
Snow monkey +12
Sharkjira + 13
Cocker Spaniel Cu-Sith +13
Border Collie Cu-Sith +13
Shar-Peï Cu-Sith +13
Inferox +13
Fairy +15
Gallop
Kojika +2
Geco +2
Geca +2
Tsuru +3
Seahorse +3
Boto Piñata +3
Iguano +4
Iguana +4
Ezoshika +4
Jellyfish +5
Tengu +6
Rupicola +6
Surgeonfish +7
Parakeet +8
Sown Goku +10
Pyro Shark +13
Ghost Shark +13
Monster Eggs +13
Sea Otter Piñata +14
Trot
Puercoespína +2
Puercoespín +2
Kitsune +2
Krill +2
Koi +3
Cardinal +3
Manatee Piñata +3
Kyuubi no kitsune +4
Mapache +4
Mapacha +4
Crab +4
Kingyo +6
Walrus Piñata +6
Shika +8
Panda +8
Kyūbiko +9
Hedgehog +10
Parrot +10
Great White Shark +13
Big Daddy +13
Chihuahua Cu-Sith +13
Scottish Terrier Cu-Sith +13
Giant squid +14
Raven +14
Emperor Frosty +15
Speed
Muerciélaga +2
Muerciélago +2
Shiba Inu +2
Caracara +2
Itachi +3
Ostrich +3
Banken +4
Macaw +5
Jaguar +6
Kamaitachi +6
Magnificent frigatebird +7
Tsuchinoko +8
Lionfish +8
Skoll & Hati +10
Narwhal Piñata +10
Crabby + 11
Batinson +11
Wild Boar +12
Reef Shark +13
Cookiecutter Shark +13
Cat-Siths +13
Cath Palug +14
Stamina
Neko +2
Ajolota +2
Ajolote +2
Beluga Piñata +2
Grizzly +3
Tora +3
Nekomata +4
Curl-crested Aracari +5
Alebrijow +6
Komainu +6
Cowfish +6
Stuffed unicorn +6
Cachalot Piñata +7
Ryū +8
Axolotl dragon +10
Sea turtle +10
Ermine +10
Akita Inu +11
Lizanion +12
Magmajira +13
Siberian Husky Cu-Sith +13
Great Dane Cu-Sith +13
Saint Bernard Cu-Sith +13
Phoenix Piñata +14
Ow +15