Jerk Ass: Big Bob, The Jolly-Olly Man, Oskar Kokoshka, Jaime O, Nick Vermicelli, and Wolfgang consistently. A lot of the kids (Helga, Harold, Sid, Stinky, Rhonda, etc.) have their moments frequently as well. There are also several one-time characters that are like this(Doug, the Jolly-Olly Man's father, Frankie G, Ludwig, The Sewer King, etc)
Jerk Jock: Wolfgang, and most of the fifth-graders.
Karma Houdini: Most of the bullies never get punished by adults. A lot of people do some very rude/mean things and get away with it. Especially the older kids like Wolfgang. Only once do we see a couple of them get told off or called out on their actions...and they were one-shot characters with no names.
Tearjerkinglydeconstructed in "Helga and the Nanny". When Miriam gets a new job, Helga’s parents hire a nanny named Inga to look after Helga. However, while Inga actually cares for Helga, Helga feels as though Inga is trying to control her life, especially when Inga calls her out on her bad behavior, telling her that sewing is a sane way to canalize anger. Finally, Helga loses her paitence completely, and frames Inga for theft...only to be surprised when the other kids call her out. Inga is fired and has to return to her country. When Helga finds her, her guilt gets the better of her, and she confesses to the framing, only for Inga to tell her she knew it all the time. Helga is confused as to why she is getting off scot-free, until Inga explains that she is not: at the end of the day, Helga is nothing but an angry and sad kid, who damages those who care about her, and because of that she cannot be happy. The show ends with Helga in her unhappy home, sadly sewing something, realizing that her actions have cost her a happy home and someone who cares about her.
Eugene was facing a possible expulsion for pulling a fire alarm, but when it was revealed that Curly framed him, Curly never faced such a punishment, even after pulling the alarm again right in front of Principal Wartz. Or if anything, he was never sent to seek the psychiatric help the he so obviously needed.
Know Your Vines: The early camping episode ended with Big Bob Pataki rushing through a small bush to get back to his campsite after a disastrous hike, cutting off Arnold telling them to go around it. This bush was poison ivy, causing him more misery.
Rhonda sometimes gets this in episodes that focus on her. The episode "Rhonda Goes Broke" comes to mind.
Harold, anytime "Ridi Pagliaccio" acts as his Leif Motif
Laser-Guided Karma: In "Phoebe's Little Problem", Harold is the only one who still makes fun of Phoebe after she vindicates herself at the end of the episode (as well as probably being the one who made fun of her the most throughout the episode). He then wets his pants at the end of the episode, becoming the new Butt Monkey.
Big Bob always falls victim to this.
Last Day to Live: Helga gets kissed by a monkey and believes that she has contracted monkeynucleosis. She believes she's going to die and soon finds herself on her death bed, her health failing. She decides to tell everyone exactly what she really thinks of them and give them her stuff. Eventually, she brings Arnold into her room to tell him her true feelings, but before she can her friend barges in and reveals that monkeynucleosis doesn't really exist and that Helga won't die after all. And thus, Helga's secret lives on.
In the pilot (later remade into the episode "24 Hours To Live"), when Arnold thought he was going to be beaten to a pulp by Harold the next day, Helga followed him around and shouted "_hours, _minutes, and _seconds until you DIE!" to illustrate the point.
Also in the episode Grandpa's Birthday, where Grandpa believes he will die the moment he turns 81, according to family history.
Left Hanging: Will Arnold ever find his parents? We'll never know.
Lethal Chef: Arnold's grandma is implied to be this at least twice. According to Grandpa in "Eating Contest," one of her recipes even included socks.
Leit Motif: Violins tend to start playing whenever Helga's "soft side" is shown. Also, there's a "Helga loves Arnold" theme that plays sometimes when she's goes on one of her love rants or has a special moment with him. This theme is usually played on violin, but there are some interesting variations. For instance, it's a bass riff in the Disney Acid Sequence of "Helga's Love Potion", and it gets a full orchestral upgrade in the movie when she kisses Arnold.
Several other characters have Leit Motif as well. Phoebe has one that is apparently, interestingly enough, a faster and slightly modified version of Helga's theme.
Oskar Kokoshka has a theme as well, and is apparently the only one of the boarders with a theme of their own. It plays particularly when he's being especially sneaky.
Grandpa Phil has a theme as well, that features heavily whenever he has a flashback to his youth.
Long-Lost Relative: Mai Hyunh, Mr. Hyunh's daughter. They were separated during the chaos of the Vietnam War. The Christmas Episode centered around Arnold finding his daughter and reuniting them for Christmas.
Lots of Luggage: This one's a stretch, but an episode had Big Bob Pataki take his daughter Helga and her friend Phoebe out camping with a lot of high-tech camping equipment because he was testing out the merchandise for his company. It turns out to be all junk that he later throws away angrily while Arnold and Gerald use the skills they learned from Arnold's Grandpa earlier to get back to their campsite.
Love Dodecahedron: Weird Cousin and Arnold visits Arnie make one of these where All Love Is Unrequited. With both towns combined, we have a chain: Arnold->Lila->Arnie->Lulu->Arnold->Hilda->Arnie->Helga->Arnold. Graphically, this would resemble a figure-eight.
However, one must also consider that "Arnold Vists Arnie" is All Just a Dream, so it is unlikely that Lulu and Hilda actually exist, and they are more likely Arnold's subconscious representations of Lila and Helga.
Not that the actual shape of love between the non-dream characters is much better. At it's most convoluted it's around this: Brainy has a crush on Helga, who has a crush on Arnold, who has a crush on Lila, who has a crush on Arnie, who got a crush on Helga instead.
Love Makes You Crazy: Helga, Brainy, Arnold, the whole series centers on unrequited love that indeed makes the characters crazy.
Amusingly, love is also the only thing that consistently makes Arnold act in ways his conscience would normally disagree with, especially later on in the show when he's almost exclusively squeaky clean.
Loves My Alter Ego: Arnold, Helga, and "Cecile" make this a very odd incarnation of the trope.
Loving a Shadow: In Helga's Masquerade, Helga tries to be more like Lila so Arnold would like her more. He does, but only because of thinking of her as Lila.
Mad Libs Catch Phrase: Phoebe's "[verb]-ing!" in response to Helga asking her to do things. Most often it's "forgetting!" whenever she and Helga have a conversation that "never happened." Helga borrows it in "Phoebe Breaks a Leg" as part of the Hourglass Plot.
Magic Realism: Pigeon Man, as well as Eugene's excessively bad luck.
Maybe Magic, Maybe Mundane: In many episodes, the characters will investigate an urban legend. Almost every time something related to the tale comes up, a logical explanation for it is provided seconds later. Toward the end of the episode, they decide it was all just a story and don't believe it, only to have the episode conclude with a heavy implication or just blatant proof that the legend was true.
Mean Character, Nice Actor: In universe - Arnold of all people is chosen to play the "black hearted villain" for the titular school play in "Eugene, Eugene." But so nice is Arnold that he can't go through with the director's ultra-cynical rewrite of the ending in which his character wins and the hero dies totally crushed because it's just too cruel, so he helps scheme to change the ending back into its original Happy Ending.
The Movie: Was a bomb. Was actually quite profitable. In fact, according to box office revenue, the film made $15 million on just a $3 million budget. Sure, it wasn't a box office smash hit, but it made a profit of $12 million. Now with the critics, yes, it did bomb.
One little detail, though: the film's promotional budget was around $13 million. Yep, that's over 4 times the amount it cost them to make the movie itself. That means that overall, there was a net loss of $1 million.
Mythology Gag: In the episode "Monkey Business" Helga gives away her sousaphone when she thinks she's gonna die from monkeynucleosis. In the original Arnold claymation shorts, she played a sousaphone in Arnold's band.
Ernie, the construction worker that lives at the Sunset Arms. He's so short that he can fit into a briefcase (as demonstrated in the "Fighting Families" episode). Played with in "Ernie In Love," in which he falls for the tall and Lovely fashion model, Lola. He also takes a bit too much pride in his work - demolishing skyscrapers and other tall buildings.
Helga also calls Arnold this in "Phoebe Takes the Fall".
Never Say "Die": Mostly averted, as several episodes featured death as a plot device (ie: Dino faked his death in attempt to sell more records, Grandpa spent an episode thinking he was about to die, Sid thought he murdered Principal Wartz, and a few other examples). However, when they make allusions to the afterlife, they sometimes mention heaven by name, but always use a euphemism in place of the word "hell," leading to quotes like this:
Grandpa:(after thinking he died) Well, that's it, I must be in heaven. Oh no! Oskar's here! This must be the other place!
However, he does say "die" after finding out he's not going to kick the bucket for another 10 years.
"HEY ARNOLD! 24 HOURS UNTIL YOU DIIIEEE!!!"
Lampshaded in the episode "Old Iron Man" where, while under the impression that they're going to die in the water, Phil and his friend Jimmy Kafka exchange various expressions for death, such as "going to Davy Jones' locker" and "waking up to the deep sleep". Phil comes up with "The Last Tango in Paris" which Kafka criticizes for "not being a euphemism for dying".
Never Trust a Trailer: The trailers for The Movie would make you think it was primarily about a block party concert to save the neighborhood. The block party in question is only merely a very brief scene near the beginning of the film that is actually an abysmal failure for the main characters.
Nice Job Breaking It, Hero: Arnold does this to Lila's heart when he reveals that he doesn't like "like" her, and regrets it afterwards because he realizes that he does genuinely like her after the fact.
The whole Arnold and Lila thing would have never happened in the first place if Helga didn't write "Arnold loves Lila" on a wall in an alley. (She originally wrote her own name in place of Lila's but quickly changed it as a group of girls were approaching.) Lila seeing the message resulted in her being interested in Arnold and later Arnold being interested in Lila, much to Helga's frustration.
The Nicknamer: Helga, who has a name for everybody: "Football Head", "Hair Boy", and "Arnoldo" (among many others) for Arnold; "Tall Hair Boy" and "Geraldo" for Gerald; "Pink Boy" for Harold; "Princess" and "Rhondaloid" for Rhonda; "Stinko" for Stinky; "Little Miss Perfect" for Lila. Even her best friend Phoebe isn't immune: Helga usually calls her "Pheebs", but at least that one's pretty common.
No Accounting for Taste: For some unexplained reason, Lila like likes Arnold's weird cousin Arnie. This also applies to Suzie's marriage to Oskar.
Noodle Implements: "Phoebe Skips" ends with Helga and Phoebe scheming to get back at the sixth grade girls who humiliated Phoebe during the episode. We never find out what the two of them ended up doing with all this stuff, but it was probably nasty.
Helga: "Alright Pheebs, looks like I'm gonna be needing a few things. A box of thumbtacks, a ball of string and a watermelon..."
Arnold, big time. This is actually subverted in a couple of later episodes where it is implied that his subconscious knows that Helga is in love with him and is trying to get him to figure it out. Justified because he might be a little reluctant and uncomfortable being involved with someone who has given so much grief over the years...and does it right after being nice to him. He at least, however, generally shows that he knows she is a much nicer person that she lets on.
Helga, meanwhile, seems totally oblivious to Brainy's crush on her - even after he tries to give her a ring.
Oddball Doppelganger: Arnold's cousin Arnie. Later in the series, an episode shows there is a whole group of odd counterparts to every other character in the series. Well, maybe.
Odd Friendship: Rhonda and Nadine. Rhonda is a stuck-up primadonna and is into fashion, which Nadine isn't. Nadine is much more down to earth and is fascinated with bugs and arachnids, which gross Rhonda out. But despite this, they've been best friends since they were four years old.
Offscreen Teleportation: Being offscreen most of the time, it's pretty easy for Brainy to do this. Lampshaded in several episodes where he shows up behind Helga in the most improbable places, such as on Elk Island, the Supposedly haunted train, inside a tree, and in an Alley that Helga made sure was empty which got Lampshaded:
Helga: Look, Brainy, this is just weird. How is it that you're standing behind me again? How did you get in this little arch? Were you waiting for me to come to this alley? What's your deal?
Onee-sama: In a rare Western example of this trope, Lila quickly develops this dynamic with Olga due to their similarities. To the point where Helga, who usually can't stand Olga, actually becomes jealous, proving Aww, Look! They Really Do Love Each Other.
Helga somehow convinces Arnold to do this with her to make Lila jealous.
Also done when Helga "dates" Stinky in an attempt to make Arnold jealous.
The Other Darrin: Played straight with Arnold who had five different voice actors (J.D. Daniels, Toran Caudell, Phillip Van Dyke, Spencer Klein, and Alex D. Linz), but averted surprisingly well with Gerald who had an episode about his voice changing just so they could continue to use his voice actor Jamil Walker Smith after his voice changed.
Out of Focus: One of the things that went wrong with The Movie, all the main cast besides Arnold, Gerald, and Helga.
Panty Shot: Helga in "Downtown As Fruits". There are upskirt shots of her white panties under her dress and milk carton costume when she comes on stage to stall, and inform the audience of the missing fruits in the play (Arnold as a banana, Gerald as a strawberry), when she gets stuck after being sent flying across the stage from Arnold and Gerald's entrance and when she's trying to free her own head from the hole of the carton.
Parental Bonus: "Oscar Kokoshka" was also the name of a real life modernist artist.
Parental Favoritism: Strong in the Pataki family; Olga is praised for her accomplishments and is given more attention than Helga, which causes a one-sided strain between the two. Olga actually wished she could be like Helga, without all the fussing and attention.
Passive-Aggressive Kombat: Mr. Simmons' boyfriend Peter and Mr. Simmons' mother indulge in a bit of this during the Thanksgiving episode.
Poisonous Friend: Helga, at times. Though she often means well in situations where she is made to help others, her idea of the right way to do something is a little... skewed. In episodes where she gives advice or puts down a plan of action, chances are that plan will be harsh, immoral, and ultimately not helpful. She lampshades it in "Deconstructing Arnold" while explaining to Arnold why he is the right person to give advice and she... isn't.
Helga: (cheerfully) Look at me! I'm terrible at helping others. I'm the worst! I have no moral conscience whatsoever.
Eugene's optimism is almost frightening, given his bad luck.
Also Lila, who is devistatingly poor and has a Missing Mom, but always smiles and acts polite. This makes sense when you realize that Lila is now taking on the responsibility of caring for her father all by herself at the age of nine. This is already a tough task as an adult, but as a child it has to be even more difficult. Also considering that she doesn't want her father to worry about any of this forces Lila to smile and keep her real feelings tucked away. Much like in real life, Lila is an excellent example of what can happen if a child is burdened with a heavy load at such a young age considering her repressed dark side. See Stepford Smiler below.
Posthumous Character: Phil's father and grandfather are both implied to be long dead, yet both are referred to in many of Grandpa's stories and appear in flashbacks.
There was originally a tenant at the boarding house named Lana, but she disappeared after season 1. There was supposed to be a subplot with her falling in love with Arnold; unsurprisingly, Nick shot down that idea in a hurry, so Lana was removed as a result.
As well as the other tenant Mr. Smith, who was put on a helicopter.