When the animals have to promptly pose as non-sentient, they each give their species calls:
Freddy: Ferret, ferret!
Bionic Janitor! Bionic Janitor!
All of "Buyers Beware":
Freddy pretends to be a mutating antelope, having Peck sprout of his costume pretending to be a second head and shooting lasers out of his eyes, after which he explains that the lasers were caused by him drinking radioactive waste. Pig tries to debunk the idea of radiation-induced superpowers by drinking it himself and ends up turning into the Incredible Hulk, complete with the US Army attacking him. The narrator then drinks it and shrinks.
"Why Army always hound Pig? Pig smash! Pig smash you all!"
You see, the barn has a teeny, tiny PORTAL TO THE UNDERWORLD!
In "Home Sweet Hole", Pip moves in with Freddy and Peck after his hole is destroyed. When it seems like a good place to live, the two suddenly start arguing Like an Old Married Couple in Italian.
From "Fowl Play":
Gopher: Wait, if you're the good cop and he's the bad cop, then who's he? *points at pig*
Pig: I'm the funny cop. Pull my finger!
In "The Barnyard Games", Peck tries to convince Freddie that he's not accident prone by safely jumping over Freddie's hat he threw on the ground. He gets struck by a meteor.
This happens again in the first round, after being crushed by two haybails.
While trying to prank Bessie, Otis puts Freddie and Peck in charge of pulling a rope, but every time he says "afternoon," they pull the rope too soon and dump it on top of Otis three times, the last time because "he was thinking it."
Freddie: I like pulling the rope, it makes rocks fall!
In "Hypno-A-Go-Go," the Viewer Mail.
The second part is especially funny because it happens just as a hypnotized Otis is about to throw what looks like the Farmer into the butter maker.
Freddy attempts to read Otis' letter saying he's leaving the barnyard, but of course can't read so it comes out as "Donkey oil pine nut moose, confetti yippity-yap" and so forth.
In "The Farmer Takes a Woman", the animals go on a dating site to find the farmer a date and they all take part in typing a fake description of him. The result...
Otis: Herfnarshsnerfneurs7#glarb. Man I wish we had fingers!
So Pip types a better description: "Wealthy, agricultural mogul with body of a Teutonic god wants to share his heart with you", and the animals photoshop his face on a (live-action) muscleman's body for a "recent photo". And a bit later...
Freddie: You can't expect a plate of chicken to drop out of nowhere (nothing happens) I'M WAITING!!!
And a while later, he's still waiting.
In "Abby and Veronica", Otis, Pig, Pip, Freddy, and Peck catcalling Veronica and later, Abby. Otis's line is especially funny: "A lady!"
From the episode "Sun Cow", Otis and a blind elder cow must compete in a round of blindfolded combat. Cue them dancing around waving their sticks like idiots and hitting random things like trees and other people.
One of the alien invaders from the Aliens TV special is a Captain Kirk expy.
"Ask Dr. Pig" from the episode with the duck, Crebs. He [Crebs] even interrupts the middle of the segment and attacks pig!
When the barn animals think the farmer has decided to eat them, Otis and the gang run off to hide somewhere. While their hiding spot looks perfect, Otis ends up fearing that the rest of the barn won't be safe, so he tries to motivate everyone else to leave and save them. When that fails, he does this:
Otis: (in a bad falsetto while gripping his own neck) Get in the cart or the cow gets it. (everyone starts begging the "kidnapper" not to hurt Otis, except Abby who just rolls her eyes)