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.
Base Breaker: Gaz isn't universally despised, but the people who do dislike her dislike her quite strongly. To her fans, she's a funny, tough, scary but badass force of nature everyone on the show thinks twice about crossing. To her detractors, she's a selfish, cruel, vindictive hypocrite who lashes out without responsibility or remorse.
In "The Girl Who Cried Gnome" there was an incident where Zim let loose a "Robot Gopher" to get rid of a girl scout that had wandered up to his front door selling cookies. After burrowing underground and trapping the girl's foot in a hole the Gopher proceeds to dance with her and then flew away into a swirling vortex leading to another dimension.
Zim: "I don't remember programming that..."
In "Gaz, Taster of Pork," Dib goes to MacMeaties and talks with a hobo. But wait for it... he spontaneously kidnaps a poor guy eating there. We never see the hobo again.
And then there's that Ham Demon in the Tak episode. Shows up for one scene, fights Zim, never mentioned again.
The infamous dancing skeletons in top hats from the Halloween special.
Professor Membrane, so very much. He has a switch that restores power to the entire earth, flies and shoots energy blasts when excited, builds and refuses to use a perpetual energy device on a whim, etc.
Dib and Zim piloting Mercury and Mars, respectively, as ships in a space battle for the continued existence of mankind.
Cult Classic: It's been cancelled 10 years and still gets decent money from merchandise. Not to mention the huge, still-alive fanbase.
Just to drive the point home: to commemorate the 10th anniversary of the show's debut, fans recently held the first "Invader Con", which is exactly what it sounds like - an entire convention dedicated to the show featuring several of the voice actors as guests.
Darkness-Induced Audience Apathy: Given how massively bleak the world is, and the constant failures of the protagonists, it's possible Jhonen invoked this on purpose.
Designated Villain: Sizz-lorr, who only does his job and retrieves Zim to give him the punishment he deserves for fleeing the Irken authorities. What the show tries to do to make him look like the Big Bad of the episode is his Evil Laugh and his super accuracy of keeping Zim from finding any escape routes.
Tak's only in one episode (and an unrecovered script), but in fanfic she often gets billing as a main character. She is the only significant female Irken. Counselor Dwicky from "Vindicated" is similarly popular.
GIR is popular to an absolutely absurd degree. To go by merchandising, not only is he the most popular character in the show, his popularity might actually exceed the show's, as people have been known to buy GIR memorabilia despite having never seen Invader Zim. As the creators themselves put it, "America loves GIR!"
Keef and Gretchen also have become rather popular. Albeit not the level of Tak and Dwicky.
The show itself could be thought of as one on a larger scale. It got canned ten years ago and only lasted for a season and a half, but it's one of the most popular Nickelodeon shows of all time.
Fandom Berserk Button: The issue of "Vort" v "Vortians". Technically, only the former is in the actual show. "Vortian" did show up in an unproduced script, but it's possible that was just a mistake that was overlooked. It's also possible that both are considered fine. But there are some purists on the issue of "Vort" being the only acceptable term.
Fanon: Fan depictions of Professor Membrane's wife (if he has one) usually depict her as looking like Gaz to explain why Dib resembles their father but Gaz does not.
Sort of gets a look in the series with Dib and Tak, as he was stated to like her (though he may have meant platonically) and offered her some of the valentines given to him by Gretchen. She, of course, is a ruthless and competent female Irken who shares his hatred for Zim (if not exceeds it) but is planning to conquer/destroy Earth herself as part of her plan to discredit Zim. However, she only showed up in one episode and Dib seemed to lose any attraction to her after discovering she is also an alien invader.
In one episode, Zim catches Dib sneaking around his house in a squirrel disguise. After Dib runs off, Zim yells a parting insult to him, calling him "Squirrel boy!" Several years later, Zim's voice actor would go on to voice the titular character in the series Squirrel Boy.
This, when you consider the internet meme involving trollface an "u jelly?"
Dib: You're just jealous... Zim: This has nothing to do with jelly!
Pretty much everything, if you've played Psychonauts. A short, goggle-wearing character voiced by Richard Steven Horvitz, dealing with a paranoid security guard, wearing goggles, enjoys setting things on fire, and screaming about meat? There's also Zim trying to justify his green skin with "a skin condition" and Psychonauts' cast featuring several green, blue, and purple-skinned characters.
Hype Backlash: As it is with everything popular, there's always a couple of people who think the show is overrated.
The most common accusation is that fans of Zim are all hyperactive "I'm so random" 12-year olds who only like the show for Gir. This certainly isn't true - there are many other older, more mature fans out there who appreciate the show for its Black Comedy, social satire and manic absurdist style. And Gir.
Internet Backdraft: And how.Operation Head Pigeons was first started on Facebook, and still remains a group as Operation Head Pigeons 2.0. Nickelodeon says that they've never seen a fanbase as hardworking as them. They've most likely triggered a DVD release, and through whatever miracle, possibly could revive the show. (Though this idea seems to be on the backburner now.)
Some of Zim's fans have the tendency to play him up as The Woobie while entirely glossing over the whole "Kill All Humans" thing. Rather than an unstable loose cannon who often fails due to a tendency to shoot himself in the foot, he's depicted as intentionally harmless. On occasion Mary Sue has been known to help Zim on his mission to conquer the Earth.
Zim's attempted murder of Dib in "Bad, Bad Rubber Piggy" does it for some. It's not really the worst behavior he's ever demonstrated, but the rapt, gleefully anticipatory look on his face as he watches the life go out of Dib's eyes — with accompanying flatline beep — is creepy on a personal level that he doesn't often reach.
Others can say when in Hobo 13, Zim sacrificed everyone on his team to advance himself to the fortress of pain.
Arguably, Gaz in "Dibship Rising", when she used a device to try wiping out all mankind.
In "Ten Minutes to Doom", with the help of Zim, she went after Dib, not because Zim was dying, but because she believed that Dib stole her video game (even though it was really Zim).
"Career Day", "Rise of the Zit-Boy", "Dark Harvest", the turkey dude's lucky neck meats in "The Sad, Sad Tale of Chickenfoot"... speaking of which, the scene where a guy's eating soggy french fries out of a tub full of dirty dishwater grosses out Jhonen himself.
"Halloween Spectacular of Spooky Doom." Just... that. *shudder* In fact, Squick is what resolves the entire episode.
Never Live It Down: Though the series-wide references to Dib as insane probably don't help it, nor the Parental Neglect, the bulk of Professor Membrane's reputation as an abusive parent probably comes from the Halloween episode, where he blithely lets Dib be sent to an insane asylum when Dib is not crazy, just suffering side-effects from using Professor Membrane's prototype trans-dimensional viewing device, something that Membrane himself admits exists.
Professor Membrane is also a frequent target. In canon he is a very neglectful and sometimes insensitive parent (what with the tendency to call his son "insane" to his face), and has at least a few jerky acts under his belt, but he does genuinely love his kids, which is most notably shown in Bad, Bad Rubber Piggy, where Dib's near-death experience causes Membrane to bring him back to life and encase him in a Giant Mecha "exoskeleton" to keep him safe. In fanon, he can be depicted as outright malicious and abusive to his children, particularly Dib.
To round out the family, Gaz is also a frequent target. In the improbable but common "Bad Future"-style fics where Zim (somehow) has the backing of the Irken army and has taken over Earth, she's very likely to have sided with the invaders out of selfishness or misanthropy. She also on occasion has her abusive tendencies towards her older brother exaggerated beyond recognition.
Torque Smacky is a weird case. In the few scenes he appeared in, he never really did anything mean to Dib, but in fanfiction he is almost always derailed into a Jerk Jock who delights in kicking around Dib and/or Zim like a football.
Mr. Dwicky from "Vindicated" gets quite a bit of flak for supposedly lying to Dib and abandoning him. While this is partially true, for the most part Dwicky was shown to be a nice guy who only "lied" to the boy so he could get to know what his real problem was. He also only went with the aliens at the end of the episode, because he, like Dib, was obsessed with Aliens as a child and couldn't resist the opportunity to travel the stars with them.
Seasonal Rot: Some consider the later episodes ("Mortos Der Soul Stealer", "Zim Eats Waffles", "Gaz, Taster of Pork" which is considered to be the weakest episode of the show.) to have suffered a significant drop in quality.
Of course, Mortos did have this memorable gem...
Zim: You can eat later, Gir.
And "The Most Horrible X-Mas Ever" turned out to be pretty awesome.
Selfish Good: Dib in the earlier episodes. All he seemed to care about was to capture Zim and prove that aliens are real, before caring about the world more.
Sending Stuff To Save The Show: The fan group Operation Head Pigeon spammed the Nickelodeon executives using phone calls and message boards. The group has prompted a Zim marathon, putting it back on Nicktoons, and a best-of DVD entitled "Operation Doom". If the DVD sells well, they hope more seasons will be greenlighted. Jhonen Vasquez has said that while he doesn't think the show will be brought back, he appreciates the group for giving hell to Nickelodeon.
Cargo Ship: Sort of an inversion. GIR seems to have sort of a little-kid crush on Gaz. It's rather subtle though. Evidence for this can be found in the episodes "Bloaty's Pizza Hog" and "Tak, the Hideous New Girl".
Same thing could be said for Zim/Gir as well. Yes, it exists.
Foe Yay Shipping: The most common (by far) pairing you will find is Zim/Dib. For many it's pure squick, but it's undeniably popular. Dib/Tak and Zim/Tak aren't too unpopular either, presumably for many of the same reasons that Zim/Dib is popular.
Idiosyncratic Ship Naming: Takes the form "_A_R" (for "X and Y romance"), with the blanks filled in by the first letters of the characters' names (e.g. ZADR for "Zim and Dib romance", or ZAGR for "Zim and Gaz Romance"). Made somewhat problematic by the fact that several characters' names start with the same letter.
"Speaking of Dib, Iím shocked at just how many girls like dressing up as Dib. Seriously, I saw a dozen female Dibs this weekend, and barely one male one. It seems like Dib has almost become some kind of butch-girl icon. I think thatís pretty cool."
Villain Decay: Whether you consider Zim or Dib the villain, they both got considerably worse at their tasks as the series advanced; early episodes would usually have one best the other, while later many ended with them both of them enduring some terrible trauma.
Keef, if you think about it. Nobody likes him, but he's one of the only nice kids in Skool.
Zim can be seen as this, in a way - he is, after all, mentally ill and very unstable, and he genuinely wants to please his superiors. His manic egotism and megalomania can almost seem like a defense mechanism. This doesn't excuse him for his actions, but it is easy to understand the implications behind his behavior.