These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
It's also quite a hit in Japan. International audiences in general tend to like the show.
Growing the Beard: A Grand Day Out isn't necessarily bad, but it's far more Surrealthan the later episodes, Wallace's only invention is the rocket, and the "bad guy" is merely misunderstood and takes the form of a... robot oven. With The Wrong Trousers, Wallace became far more of a tinkerer, Gromit became the archetypal Silent Snarker and the series received a massive Animation Bump.
Which comes full circle in Project Zoo, which involves those same gameplay mechanics at one point.
Moral Event Horizon: During his malfunctioning, Preston attempts to turn Wendolene and the sheep into DOG MEAT!!! Surely that qualifies.
Lord Victor Quartermaine is willing to kill the Were-Rabbit (even though he knows it's really Wallace) just so he can marry Tottington for her fortune.
Tear Jerker: When Gromit runs away from home in Wrong Trousers and when Gromit cries over Wallace's apparently dead body in 'Were-Rabbit.
Another example is in "A Close Shave." In the last newspaper with the headline of Gromit going to jail for life that Wallace and Shaun are reading, both have tears in their eyes, as do several other sheep reading with Wallace and Shaun, as Wallace says, "Oh, Gromit," before he and the sheep look at a portrait of the two together.
Also in The Curse of the Were-Rabbit, there's Wallace's reaction to learning that he's the were-rabbit everyone's been hunting. He starts desperately trying to fix a device that might reverse the process, only to find that his brain is pretty much "rabbit mush". He ends up breaking down, sobbing "Oh Gromit, I don't want to be a giant rabbit!"
Again from The Curse of the Were-Rabbit: Victor locks Gromit in a cage while he goes after the Were-Rabbit (Wallace). Gromit desperately tries to get out for a few minutes. Then he slumps, and, even though there's no sound and we can't see his face, it's clear that he's crying.
That One Sidequest: Trying to get all the coins in Wallace & Gromit - Project Zoo? If you are, then the mere mention of the first Lava World bonus stage will make you curl up in the fetal position and whimper. Gromit has to climb a series of platforms while avoiding rolling barrelsthrown by a gorilla. The problem? The GameCube version of the game contains a glitch where most of the barrels are invisible to the player. Barrels that result in instant death if touched. Oh, and did we mention you're on a time limit? Getting the coins from this level requires truly psychic guesswork and timing and the reward? Short clips from "The Wrong Trousers" and a single preview from the 'Cracking Contraptions' series.
Tough Act to Follow: The Wrong Trousers is a mild example, still near-universally considered the best in the series, but with the others all following close behind. A much bigger example is the short's villain, Feathers McGraw, who's considered such an iconic character that none of the succeeding villains are anywhere near as well-remembered.
The Woobie: Gromit, particularly in Wrong Trousers. Fluffles also qualifies up until the end of Loaf and Death.
Fluffles is probably the most Woobie character in the franchise. Piella hit her and you could literally hear her yelping in pain! It is also vaguely implied that she has suffered this abuse for quite a while
Wallace at the end of A Close Shave. After everything that they had gone through up to that point to earn their happy ending, Wallace finds out that Wendolene is allergic to cheese and can't stand it at all.