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
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
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
Okuri Inu +6
Gouldian finch +6
Namazu +10
Fox + 10
Snow monkey +12
Cocker Spaniel Cu-Sith +13
Border Collie Cu-Sith +13
Shar-Peï Cu-Sith +13
Fairy +15
Gallop
Kojika +2
Geco +2
Geca +2
Tsuru +3
Seahorse +3
Iguano +4
Iguana +4
Ezoshika +4
Jellyfish +5
Tengu +6
Rupicola +6
Surgeonfish +7
Parakeet +8
Sown Goku +10
Monster Eggs +13
Trot
Puercoespína +2
Puercoespín +2
Kitsune +2
Krill +2
Koi +3
Cardinal +3
Kyuubi no kitsune +4
Mapache +4
Mapacha +4
Crab +4
Kingyo +6
Shika +8
Panda +8
Kyūbiko +9
Hedgehog +10
Parrot +10
Chihuahua Cu-Sith +13
Scottish Terrier Cu-Sith +13
Giant squid +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
Wild Boar +12
Cat-Siths +13
Cath Palug +14
Stamina
Neko +2
Ajolota +2
Ajolote +2
Grizzly +3
Tora +3
Nekomata +4
Curl-crested Aracari +5
Alebrijow +6
Komainu +6
Cowfish +6
Stuffed unicorn +6
Ryū +8
Axolotl dragon +10
Sea turtle +10
Ermine +10
Akita Inu +11
Lizanion +12
Siberian Husky Cu-Sith +13
Great Dane Cu-Sith +13
Saint Bernard Cu-Sith +13
Phoenix Piñata +14
Ow +15