My personal ranking:
A Rank (useful in a wide variety of events, or for earning CP.)
Blackbeard Randy (
Tiger) - Massive attack, great for single-target battles, and Claw Cyclone makes for a decent conal option.
Bouncing Bertha (
Chapuli) - Tegmina Buffet has the largest fTP as far as Ready moves are concerned; can self-skillchain with Sensilla Blades.
Vivacious Vickie (
Boar) - Sweeping Gouge is 2 hits with replicating fTP, and the DEF-25% additional effect is amazing. Zealous Snort turns her into a formidable tank.
Warlike Patrick (
Lizard) - Nice self-skillchain options, and can MB on its own Liquefaction or Fusion. Opens Impaction skillchain for CP parties/Genbu/Ironside/etc.
B Rank (edged out by superior options, but still have niche uses.)
Headbreaker Ken (
Fly) - Cursed Sphere is stronger than Fireball, making it the best magical AoE cleave pet for anything that doesn't resist Darkness.
Swooping Zhivago (
Tulfaire) - The AegisBird™. Takes -50% damage from slashing attacks and fire/wind. Swooping Frenzy inflicts DEF-25% and MDB-25.
Alluring Honey (
Snapweed) - Takes -50% damage from Thunder, Wind and Darkness attacks, which can be useful for a number of battlefields.
Rhyming Shizuna (
Sheep) - The pet with the highest attack BUT requires spending 2 charges on Rage. Sheep Song can be beneficial in certain SKCNM battles.
Amiable Roche (
Jagil) - Insanely resistant to anything Water-based, and has a really nice attack modifier. Decent choice for the Leviathan battlefield.
Hurler Percival (
Beetle) - Unfortunately it's a PLD (so no Fencer TP Bonus) but has decent attack and type coverage. Useful for Plouton.
Fleet Reinhard (
Raptor) - Good attack power and can cover Blunt and Slashing weaknesses, making it another candidate for Plouton.
Gussy Hachirobe (
Spider) - Also has good attack power, but sadly Sickle Slash is a crit-based move so it'll never do more than 8-10k damage.
Pondering Peter (
Rabbit) - Large radius AoE with Whirl Claws, and an AoE healing option with Wild Carrot. Foot Kick suffers from being crit-based.
Threestar Lynn (
Ladybug) - Treasure Hunter companion. Noisome Powder is a good AoE Dynamis JA proc that inflicts Atk-10%.
Sharpwit Hermes (
Mandragora) - Deals blunt, piercing, and slashing damage. Useful for fighting Warder of Temperance and Caturae monsters.
F Rank (mascots and other ill-conceived monsters.)
Brave Hero Glenn (
Toad) - No ready moves.
Brainy Waluis (
Funguar) - Frogkick ignores target's defense, which is rarely - if ever - useful. If it bypassed PDT, that'd be another story.
Herald Henry/Aged Angus (
Crab) - Bubble Curtain's MDT-50% is the main selling point, but aside from tanking stuff solo without trusts, not terribly useful.
Suspicious Alice (
Eft) - Geist Wall and Cyclotail are mildly useful, but this pet has mediocre stats and no notable strengths.
Anklebiter Jedd (
Diremite) - Spinning Top is powerful and has good range, but the Diremite is a Dark Knight which means no Fencer. Decent for farming fodder.
Choral Leera (
Colibri) - Low attack and not a WAR pet. Pecking Flurry is its only move (hits 4 times). Not bad for stuff like Nosoi/Vedrfolnir, but Beetle is superior.
Cursed Annabelle (
Antlion) - Low attack. Sandpit might have some untapped applications, but generally an underwhelming pet.
Submerged Iyo (
Apkallu) - Wing Slap<->Beak lunge can self-skillchain. Incredible for tanking Water magic spells, but offensively disappointing.
Generous Arthur (
Slug) - Purulent/Corrosive Ooze have excellent support role potential, but outside of Unleash the charge-requirements are too steep for general use.
Fluffy Bredo (
Acuex) - Takes -75% damage from blunt attacks, which is impressive. Has high attack, but no physical Ready moves.
Left-Handed Yoko (
Mosquito) - Conal HP absorb/plague with Infected Leech (ew) which seems cool at first, but Gloom Spray is very weak considering the 2-charge cost so this pet kills very slowly.
To look at it from a different perspective, based on events:
High-Tier Merit Battlefield: Tiger (most), Boar (most), Snapweed (Fenrir/Diabolos/Promathia/Garuda/Ramuh), Tulfaire (solo Ark Angels), Jagil (Leviathan)
Domain Invasion: Chapuli, Tiger (Azi Dahaka/Naga Raja; use Claw Cyclone while they're flying), Lizard (Fireball when adds spawn)
Delve: Tiger (most), Beetle (Hakawai), Snapweed (Cailimh)
Incursion: Chapuli
Sinister Reign: Tiger
Vagary:
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Deathborne Gate [Fomors]: Chapuli
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Brash Gate [Amorphs]: Chapuli
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Duskbrood Gate [Elementals]: Lizard (until Umbril NM), Fly (until Pixie NM), Tiger (Putraxia)
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Perfidien: Antlion+Ladybug (4-step skillchain sequence), Lizard (Brain Crush->Tail Blow->Fireball MB kills on Ice Elementals)
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Plouton: Antlion+Ladybug (6-step skillchain sequence), Spider+Ladybug (Sickle Slash->Spiral Spin->Acid Spray non-lethal MB), Lizard (Fireball kills on Elementals), Beetle (Plouton below 50% HP)
Gaes Fete:
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Zi'tah: Tiger (most), Boar (most), Beetle (Nosoi), Chapuli (Pixies), Lizard/Slug/Acuex (Wrathare)
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Ru'aun: Tiger (most), Boar (most), Mandragora (Bia, Khon, Khun, Ma, Met, Ruea), Chapuli (Kammavaca, Vir'ava), Lizard (Impaction on Genbu for Thunder MBs)
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Reisenjima: Tiger (most), Chapuli (Mauve-wristed Gomberry, Boobrie adds during Erinys battle), Beetle (Zduhac), Lizard (Impaction on Ironside for Thunder MBs)
Your choice of pet may vary depending on the party setup, but these are my baseline go-to pets.