It depends on the char.
Funny i am not the only person, that prefers the Battleship for DH. 
Necro is running Liz Moore and would also sometimes run a flying Skull or walking hand, as soon as it drops.
Monk is running Rocky (though I dont like it) waiting for the Emeralds Dragon to appear next season.
Crusader is running Galthrak the Unhinged or Season 21 pet and waits for The Unicorn or The mimik to drop (as one of my favourites)
WD is running cute Old Growth, waiting for Lady Morthanlu drop (too bad I cannot get the chicken).
Barb is running the Bumble, waiting for The Butcher to appear again in Januar.
Wizard is running the season pet from last season, waiting for drop of Queen of the Succubi or to also sometimes run the actual season pet.
I think Mimik could be my favorite, but there are other candidats, too and it depends on their movement.
I like to show off the Diablo pet. And I always put on my CE angelic wings when pub-bountying.

Liv Moore by far. Sadly I donāt have it on PC, only on my Xbox account. On PC I use Lamb.
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mothership from starcraft is a favourite. also halfgolem kinda suits necros =)
and the steampunk pet from season 21
but i usually choose something to look best with my transmog and its color
A few favorites like the orphan for Crusader, the battleship, the angelic goblin and a few more oh the mini butcher as well albeit noisy.
My favorite is Dominionās Revenge. My monk runs with it ever since it was introduced. My DH runs with the Queen of the Succubi. All of my mules run with different pets; I try to match the pet with the class as close as I can:
Unihorn for my Crusaders, Old Growth for my WD, Haunting Hannah for my Necromancer, Blaze for my Wizards etcā¦
I also even try to match their colors with my toonās outfits as much as I canā¦Iām very into coordination! 
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You might find this helpful: List of all Wings, Cosmetics, Pets, and Promos
Thereās 19 pets from pet goblins, 11 from seasons, 2 from RoS/Necro DLC, 2 from the Darkening of Tristram event, one from a unique mob (Liv Moore from Ravi Lilywhite). Then thereās some more from Blizzcon as well as from cross-promotions with other Blizzard games.
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Queen of the Succubi and Emerald Serpent
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Miss January 1987ā¦ she was my favorite pet back in the dayā¦
I just saw Grunkk. I WANT A GRUNKK NOW!!! He is a baby Siegebreaker WITH DIAPERS!!! Bwahahahahahahahahahahahaha.
I use the mimic on my DHās and the bumble on my WDās.
I only have like 5, or 10 or some small number. I wouldnāt have used them if it wasnāt for them serving a purpose.
I quite like the Butcher one, just for nostalgia. And I would have liked the Battlecruiser for the same reason if it didnāt stand out so much. O and the evil unicorn horse thing is cool.
I always use the little Maleasance one when playing a Necromancer. He is probably my favourite.
i usually pick my pet to match my characterās aesthetic, but Lamb is my favorite. we really have no business dragging that poor kid through hell heaven.
I heard a looong time ago that if you have The Stomach as a pet and drop the Horadric Hamburger on the ground, he will actually eat it. I keep forgetting to confirm if this is true, has anyone?..
I will start saving all of them till I get that pet. That would be hilarious to see.
Iām a new player but I have SC2 so the Battle cruiser.
The Bumble, Angelic Goblin, Royal Calf, Buddy and Old Growth