One-of-Many

Easily nuke this fella using an Unholy Val'kyr and one of the Zandalari raptors;

Doom + Ascension, followed by Black Claw + Hunting Party, and whatever damaging ability after to finish it off if not dead yet.

EDIT: IF YOU'RE LOOKING FOR THE ARGUS PET TRAINER, THIS IS THE WRONG PAGE. CLICK HERE: [npc=128024]/one-of-many

I'm going to record what worked for me. I'm making no claims that these work reliably, I may have had good RNG and I didn't do a lot of testing. Breeds may have played a factor too so I will record them when I remember to do so.

Humanoid Havoc

It can be difficult to get him him to die since he has such a powerful heal. Here's what worked for me.

Squirky - 2 2 2
Hogs - 1 1 1
3rd pet - any humanoid

Squirky - bubble, immediately swap to Hogs
Hogs - Stampede, swap back to Squirky
Squirky - Stampede again. die
Hogs - headbutt, then vicious slice. This should kill him, but he's undead, so he'll survive anyway. Use any move you want to wait out the turn. He'll probably kill you, but you still have a 3rd, never used pet you can call if you need to. This may take some RNG but I tried dozens of combinations and this is the only one that ever worked!

Fierce Fliers
For this one I just used the three fastest flyers I have with swarm ( [pet-ability=706 domain=www] ). Note that Tiny Flamefly doesn't have Cocoon Strike, which is very useful for countering Antaen Cannon. Anything with a speed greater than 281 will do.

I used:
Firefly S/S - 1 2 1
Shrine Fly (lvl 23) - S/S 1 2 1
Mei Li Sparkler H/S - 1 2 1

Firefly - Cocoon Strike, Swarm, Scratch, Scratch, Cocoon Strike, Swarm (will die part way through this)
Shrine Fly - Cocoon Strike, Scratch. Mine critted here, but it should be pretty easy from here even if that doesn't happen for you.

Unstoppable Undead

WIcked Soul S/S x x 1
Blighted Squirrel P/B 1 x 2
Any other undead pet

Wicked Soul - Haunt
Blighted Squirrel - Stampede, Scratch, scratch, Opponent will die, do anything for a round until undead debuff wears off and claim victory!

Critical Critters
I used the three fastest hares/rabbits I had with Scratch, Dodge, Stampede

Alpine Hare S/S
Brown Rabbit S/S
Snowshoe Rabbit S/S

Alpine Hare - Dodge, Stampede, Scratch, Scratch, Scratch (die)
Brown Rabbit - Scratch, Dodge, Stampede, Scratch, Scratch, dodge
Last pet never even used.

Magical Mayhem

This one is a mess. I tried to find a strategy that didn't rely on Lens Flare as it only works 50% of the time, but I struggled. All I can tell you is what worked for me, but it may take you a couple of tries.

Eye of Observation - 1 2 2
Lil' Leftovers - 2 1 2
Spectral Porcupette 2 2 1

Eye of Observation This is tricky. Depending on how your Lens Flares go, you may live a lot longer or shorter than I did. Basically just use him til he dies. The rotation goes Powerball, Powerball, Lens Flare, repeat until death. From here on out our paths may differ wildly since this part is so random, but here is what I ended up doing;

Little Leftovers - Food Coma, Chew

Spectral Porcupette - Illusionary Barrier, Powerball, Powerball, Spirit Spikes, Powerball, Powerball, Illusionary Barrier, Pass, Spirit Spikes, Powerball, Powerball, Powerball, Powerball

Little Leftovers - Food Coma

The only tips I can give you here are, be careful with the Illusionary Barrier and make sure you're paying attention to the speed stat/who goes first. If you're trying to get powerball's total hit up it can be a good idea to run into the barrier on purpose, but make sure not to accidentally block your own chew from the back line with it! I think my original pass at this I did that.

Elemental Escalation
The first time I managed this, I'm sure I did it in fewer moves, but when I went back up the log it didn't even cover close to half the fight because of the ticking nature of blistering cold. I recorded the exact moves I used the second time, and this will work even if it has a few unnecessary steps. The super extra double important thing here is NOT to let blistering cold drop off. If it drops off, it has to start building up from 1 tick again. Please note it's going to look like you're not going to do enough DPS until the very end.

Nightshade Sproutling S/S (Any breed of 282 and higher should work though) 1 1 1
Crazy Carrot 2 2 1
Ominous Flame 1 x 2

Nighshade Sproutling - Blinding Poison
Crazy Carrot - Blistering cold
Ominious Flame - Forboding Curse
Crazy Carot - Blistering cold
Nightshade sproutling - Blinding Poison
Crazy Carot - Blistering cold
Ominous Flame - Forboding Curse (die)
Nightshade Sproutling - Lash, Lash, Blinding Poison
Crazy Carrot - Blistering cold, Iron bark, Iron Bark (die) (possibly 1 more iron bark needed to die)
Nightshade Sproutling - Blinding Poison, Nature's Ward

Beast Blitz
My team is very similar to Ryan's Team.
Death Adder Hatchling S/S - 1 1 2
Zandalari Kneebiter S/S - 2 2 2
Zandalari Kneebiter P/B - 2 2 2

Death Adder Hatchling: Blinding Poison
Zandalari Kneebiter: Black Claw, Hunting Party, Blood Fang, black claw die
Death Adder Hatchling: Poison Fang, Blinding Poison, Puncture Wound
2nd Zandalari Kneebiter: Black Claw, Hunting Party

Aquatic Assault
Slithershock Elver S/S 1 1 1
Yellow-Bellied Bullfrog P/P 2 1 2
Benax 2 2 1

Slithershock Elver - Dodge
Benax - Bubble, Swarm of flies
Yellow-Bellied Bullfrog - Healing Wave, Tongue Lash, Tongue Lash
Benax - Swarm of flies, Tongue Lash, Bubble, Tongue Lash, Tongue Lash
Slithershock Elver - Dodge, Deep Bite, Dive, Deep Bite, Deep Bite, Dodge, Deep Bite, Dive (Die)
Benax - Bubble

Mechanical Melee
Clockwork Gnome x x 1
Cogblade Raptor x x 1
Lil' Bling 2 x 2

Clockwork Gnome - Build Turret
Cogblade Raptor - Exposed Wounds
Lil' Bling - Inflation, Inflation (die) (win)

Method for Humanoid Havoc and One-of-Many

NO RNG! TWO PETS! TWO ABILITIES! SUPER EASY METHOD!

Killing One-of-Many is actually super easy and can be killed using just two pets with only two abilities. This makes both the achievement and daily super easy.

The two pets you will need are:

• 
  
• Corefire Imp with Wild Magic
• 
  
• Hogs with Stampede
• 
  

Strategy Steps:

1. 
   
2. 
   Corefire Imp
   
   • 
     
   • Wild Magic
   • 
     
   • Pet dies
   • 
     
   
   
3. 
   
4. 
   Hogs
   
   • 
     
   • Stampede
   • 
     
   • Stampede again!
   • 
     
   • One-of-Many dies, enjoy!
   • 
     
   
   
5.