There have been a number of running gags throughout Eddsworld's history, most notably Edd's Coca-Cola addiction and Tom's "Holy____ in a _____!" line. Here's a list of them, major or minor.
List of Major Running Gags
Edd's Coca-Cola Addiction
Edd's coke obsession has been featured in many Eddisodes and is sometimes a main plot development in a story, such as Edd's decision to massacre all of the clones in Spares. It has often saved Edd from difficult situations, like when Edd used it to fend off his future self in WTFuture. Sometimes Coke has been the inspiration behind a video, such as Sugar Sugar. Edd's addiction first appeared in Eddsworld Christmas 2005, where Edd fell asleep at his desk after drinking dozens of cans of cola. It was a can of Cola that helped him realise he was out of the world where he doesn't exist. He is seen drinking some Coca-Cola on a beach in Tord's Adventure and offers one to Tord. In Zombeh Nation, he only goes to save Matt from a zombeh infested train station because Matt has the emergency Coke, Edd then tackles Matt to the floor to get the cola back. In Hello Hellhole, Lucifer revealed Edd's personal Hell is when a drinks dispenser refuses to give him a can of Coke. Edd's heart's desire is a Cola-filled pool.
Tom's "Holy ____ in a _____!" line
Tom's holy line is a line he says where he inserts two random objects has featured in Eddsworld since Zombeh Nation, and is popular amongst fans, there is a T-shirt available with that line written on it. Not all "Holy" lines are said by Tom though, as in Zombeh Attack 2, Edd utters "holy shit!" when encountering a horde of Zombehs.
Tom's lack of eyes has been joked about by the group many times and were first featured in Zombeh Nation and his lack of eyes has been the centre of many comics. When real-life Tom was asked by a fan about the presence of his eyes in his Tomska and Cake-Bomb videos, Tom jokingly replied that the eyes were edited into the videos after filming. In Hammer & Fail (Part 2), he rubs his hands into his eyes, revealing there are eyeballs in the sockets, but they appear to be jet black and squishy.
Tom's Hatred of Christmas
Initially, Tom notes that he doesn't celebrate Christmas due to personal reasons, starting at Eddsworld Christmas Special 2005. This seasonal gag continues through the Zanta Claws Eddisodes and Xmas Day. In the Zanta Claws trilogy, he had the intentions to kill Santa Claus, only being driven to save it or encounter Zanta Claws in the first two of three Eddisodes. In Zanta Claws II, he reluctantly saves Christmas by sacrificing his most wanted gift as his hatred remained. The short, Xmas Day, describes Tom's hatred through indifference where he maintains this expression throughout the Eddisode, possibly depicted as a scar from Zanta Claws II. Zanta Claws III occurred where he is the antagonist ruining Christmas in Zanta Claws' place, ultimately stopped by the gang with Zanta and Santa's aid.
This first appeared in Edd Again, when Matt bought a giant doll of himself, but wasn't seen for a while until MovieMakers when Matt revealed he had a DVD of "Matt on Matt Action". It features quite heavily in the comics on the Eddsworld website. Despite it being quite annoying to Edd and Tom, Matt's self-obsession proved key in defeating Future Edd, despite the consequences of Matt becoming the creator and ruler of everything in existence. It also helped power the aliens' Vanity Drive.
Tord's Gun Obsession
This first appeared in Eddsworld Zombeh Attack when Tord chose a shotgun as a weapon, only to have it exchanged for a spoon and fork by Edd and Tom, he was later seen studying the guns in a gun shop. His obsession with guns has appeared many times during episodes and has been seen to own a wide range of guns which he doesn't hesitate in using, such as an AK-47 (which he is often seen with), a Desert Eagle and two Akimbo pistols (although these were owned by some of his clones) In a comic, he once shot at Edd and Tom without thinking, and he seemed to enjoy attacking his clones.
It's safe to say that this obsession is sometimes exaggerated.
Minor Running Gags
Hellucard has his signature gag, which is used at every credited appearance, and is shown when he says "Ey Hed!". The running gag started in Spares and ended in WTFuture coinciding with him killed off by Edd's future self. At the beginning of Space Face (Part 1), his corpse is seen holding a sign in space saying "Ey Head" in the intro with Paul next to him.
Front Yard Coca-Cola Can
Tom's "Drinking December" Holiday Tradition
Tom has gotten drunk around the holidays many times. He was drunk in Zanta Claws III, and got drunk in a Eddsworld Comic. To make a long story short, he promised not to get drunk, and woke up half-naked in a bush.
Coca-Cola with Bacon
Edd's Bacon Addiction
Edd's bacon addiction was first mentioned in the game Quest for Bacon and was behind the plot of Breakfast, where it was also suggested that Tord may also have a bacon addiction. The last time Edd's bacon addiction was seen was in Egg, Bacon & Edd: Breakfast Buddies when he ate his co-presenter, Bacon.
Eat at Joe's
This is a place seen in many Eddsworld episodes. This has even been seen in some of Edd's earlier (non-Eddsworld related) cartoons. Although we have never seen inside, nor have the characters ever entered it, its neon sign is seen often.