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.
Alternative Character Interpretation: Fred Wolf Series: This theory seems to have arisen for some... is Irma chasing after guys really her desperation in nabbing herself a man, or, is it an act of desperation to cover up her being a closeted lesbian?
How most fans of the 80's series treat the Red Sky seasons (though some are fond of season eight).
Funny Moments: In the 2007 movie, Splinter talking about his "stories", especially him muttering to himself about how "Cody is going to break up with Donna, I just know it."
Genius Bonus/Shown Their Work: In the 80's cartoon, the main reason for Baxter Stockman becoming more and more absent minded after his mutation as the show went on wasn't because of Flanderization, but because flies have poor memories and can be very scatterbrained in real life.
Moral Event Horizon: Shredder officially crossed this in the opening episode "Get Shredder" where he actually destroys the Channel 6 building which takes it out of commission for the rest of the series.
The Next Mutation: Venus de Milo, especially by Peter Laird.
4Kids series: Cody Jones
So Bad, It's Good: Try watching the Christmas Special, We Wish You a Turtles Christmas without laughing.
About 90% of the 80's series.
Special Effect Failure: Any live action Ninja Turtles production that wasn't a movie. "Turtle Tunes" and "We Wish You a Turtles Christmas" being the worst offenders.
Tear Jerker: In the episode "Splinter No More", it's pretty hard to see Splinter lamenting his loss of his humanity to the point of shedding tears. This becomes much Harsher in Hindsight when you remember that Splinter destroyed Shredder's Retro Mutagen Ray in order to save the turtles, but also destroying the one thing that could return him to his human form in the process.
Also in the episode "Hot Rodding Teenagers", it's tough to see Kala break down as she explains that the only reason that she and the other Neutrinos are on the run is because that they are kids who would like to have fun instead of fighting in Dimension X's infinite war, and are being hunted down for it.
The penultimate example for most fans of the original series is in the opening episode "Get Shredder" where Shredder actually destroys the Channel 6 building which takes it out of commission for the rest of the series.
They Changed It, Now It Sucks: Fans of the original Mirage Comics dish on the Fred Wolf show for "selling out". Fans of the Fred Wolf show say this about the 4Kids version and its more serious tone. Fans of the first five seasons of the 4Kids show will scream obscenities at Fast Forward due to its radical shift in tone, setting, and look.
This was Lampshaded in Turtles Forever when the Fred Wolf turtles and 4Kids turtles arrive in the 1984 (Mirage) universe. The original turtles there call the others 'sell outs' though they eventually settle their differences.
The 2012 cartoon has suffered this from fans of both the 80's and 2003 version.
Uncanny Valley: Usually a problem with the turtles in Live Action Adaptations, most notably the cheap Coming Out of their Shells Tour and We Wish You A Turtles Christmas Special. Splinter often looks even worse.
The 2007 CGI movie, on the other hand, makes the turtles look quite nice, but the human models have been accused of this.
The costumes in the Operation Blue Line promotional video — which was about a light rail service that went from Los Angeles to Long Beach, and released only in California for free — blow everything else out of the water. 
Weird Al Effect: The original comic is a spot on pastiche of Frank Miller's work circa the early to mid 80s (mostly Ronin and Daredevil); the writers being big fans. The best example is The Foot, who are based on Marvel's The Hand (get it?).
Also compared Daredevil's master (Stick) to the Turtle's master (Splinter). Both origins involve a truck of toxic waste too.
It goes much further than that. The first issue starts with a page-by-page rip off of the toxic accident that created Daredevil, up to and including the "kid saving an old man and getting blinded" bit. If it weren't for their divergent timelines or the copyright being owned by different companies, the idea was that the blind kid is Daredevil and the canister that created him also created the Turtles. Marvel actually did have the rights to the Turtles comics at one point, but they were set in a separate universe, so perhaps the Turtles are part of the Marvel multiverse and in this one, Matt Murdock never became Daredevil and just became a blind kid.
A few elements were cribbed from X-Men as well; Leonardo was Cyclops and Raphael was Wolverine; and when Krang and Shredder◊ appear together, they seem eerily similar to Mojo and Spiral◊. While the X-Men have gone onto prominence in their own right, for a while they were less known to the public than the Turtles.
The Neutrinos in their debut episode count for this considering that they were being hunted in their home Dimension X just because they actually opposed fighting in the never ending war happening there.
Kala: "You don't know what it's like.... living in a place where everyone wants to do you in....just for the crime of being young." (She starts to cry.)
And like his mistress, Kala's pet Grybyx falls under this category in his debut episode.
Tropes that pertain to the 1989 Nintendo Entertainment System video game:
Awesome Music/Ear Worm: The overworld music for Stages 1 through 5, building theme 1 and the boss music are popular choices.
Demonic Spiders: Many kinds, but possibly the worst are those jetpack laser soldiers in the final area.
Game Breaker: The scroll weapon makes quick work on most bosses. However, it's extremely rare and can be replaced by other subweapons.
Also Donatello, considering that his bo staff had absolutely absurd offensive power and range, which made most boss fights pretty trivial.
Good Bad Bug: The North American MS-DOS version has a required jump that is literally impossible to accomplish. However, you can still beat the game without cheats, if you do a sequence of extremely strange actions (although you still skip the rest of that level). If you have rope, there's a place on the overworld that you can walk, which triggers the use of the rope. Suddenly, you're using the climbing animation on the overworld. If you now enter a specific building, you will fall through the floor into a glitched level. You might end up dead, or the game asking you to insert another disk, or just trapped in a broken level, but there is a chance that the game will instead teleport you to the beginning of the next level. 
Porting Disaster: The North American PC version to say the least. The third stage cannot be completed without cheating or a very strange glitch because of a design flaw in one of the sections in the sewers. The jump is impossible to make because of the ceiling being shorter and the platform on the other side missing a block. The UK Teenage Mutant Hero Turtles corrected this by extending the platform, but retained the other design decisions that made many aspects unintentionally more difficult.
It's a Timed Mission. You are charged with defusing eight bombs under a time limit, with annoying sea life everywhere trying to inhibit your process. Failing to defuse them all before time runs out will result in a...
Non-Standard Game Over. It doesn't matter how many turtles you have left; you fail once, and it's all over. Luckily it's the only game over of its kind in this game, since according to gamers who managed to pass this level, the game gets even harder.
Tier-Induced Scrappy: Raphael, oh dear god Raphael. The main reason for this is because instead of thrusting his sai daggers forward like he should, he just sticks out his arm and spins them in place which means he has absolutely no range on his attacks as opposed to the other turtles who actually have decent range with their attacks.This means it's a good chance you'll take damage while trying to damage the enemies. This game is alreadyNintendo Hard, but using Raphael as your main character takes this Up to Eleven and falls somewhere between Self-Imposed Challenge and outright Unwinnable.
"Shredder's Suite", the song that plays throughout the first film during different scenes, usually when the Shredder or the Foot Clan are a big part of the scene. The first film in general had a nice soundtrack.
In the fourth movie, Raphael fights a small monster inside a cafe kitchen accompanied by Big City Rock's cover of "Black Betty".
Continuity Lock-Out: The fourth movie introduces Karai and shows her leading the Foot now. However, unless you're familiar with the comics or the 2003 series, you will have no clue about her history with the Foot or Shredder. This is generally not a problem, but without the proper context, her anguished "You!" when she goes out of her to attack the Turtles upon their first encounter doesn't carry as much weight and may seem out of place.
Although it sort of works, since Leonardo is clearly as confused as to who she is and why she's so ticked off as the audience would be.
From the first movie, "Turtle Power" by Partners In Kryme. Although it incorrectly cites Raph as leader, it's still a narm charming rap of the highest caliber. It would be reused in the soundtrack CD of the third movie, though that might've been because the guys in charge just didn't care. Spunkadelic's "9.95," though to a much lesser extent, is also catchy in its own right, and would be reused in the trailer for the second movie.
The second—of course—features "The Ninja Rap" by Vanilla Ice, which asks you if you've "ever seen a turtle get down" and requests all ninjas to advance forward...in other words, "GO NINJA GO NINJA GO!"
TMNT features "Shell Shock" by Gym Class Heroes, which is catchier than it had any right to be.
Hilarious in Hindsight: Any sports fan over the last decade who's read the stories on steroid use can now look back on Raph's comment on Casey's Jose Canseco bat with a whole new meaning.
It was a two-for-one sale.
Donatello gets ragged on by his brothers for trying to make the word "Acapella" sound cool. Years later Eminem finishes what Don started, by using that word and the meaning behind it in a crucial moment against Papa Doc in 8 Mile .
Internet Backdraft: Michael Bay and Aliens. It's best not to even bring those two things up together in a TMNT forum. So much that many are considering it the worst movie ever before the trailer is released.
It seems to have died down somewhat, with the assurance that the aliens part is no longer being used.
Nightmare Fuel: In the first movie The Shredder himself is made of Nightmare Fuel, he's like a faster and more agile version of Darth Vader with more pointy things.
Special Effect Failure: There was a slight decline between the suit quality between the first and second films (They still worked fine and looked rather impressive, but the executives made them dial back the details somewhat). In the first film each suit was individually made with a build and details unique to each turtle (Raphael has some tough-guy scars, Mikey is the shortest, Donatello is kind of chubby, and Leonardo is the slimmest turtle) In the second they had four (basically) identical suits with different heads for each turtle. The third film made the Turtles look like sock puppet frogs and Splinter like a retarded Chuck E. Cheese.
Other failures that don't involve the animatronics (directly or indirectly) include:
In the first movie a clearly human hand swings in front of the camera as Donatello does a spin on his skateboard, there's a moment where you can clearly see a microphone cord dangling from Raphael's shell, and another where you can see the face of Donatello's actor peeking though his mouth.
Tear Jerker: The campfire scene in the first movie is perhaps the film series' greatest tear jerker. It comes off as even more sad if you see it as how Splinter probably intended it to be: a final goodbye.
There's actually one at the beginning of the movie too where Splinter confronts Raphael about his anger and Raph breaks down in tears over his perceived failures.
They Just Didn't Care: "Turtle Power" by Partners in Kryme goes out of it's way to declare that Raphael was the leader of the Turtles. No one bothered to correct it even when it was given a full-fledged music video. The real clue is the fact that the song is basically a synopsis of the movie, so whoever wrote the song had to have seen it.
They Wasted a Perfectly Good Plot: More like wasting a hinted plot in the fourth movie. The Sequel Hookhas Karai suggesting the Turtles will have to someday face the return of the Shredder, which fans probably would've preferred to being the actual plot instead of the one we got.
Visual Effects of Awesome: The animatronic suits and puppets from the first two movies still look unbelievably good even today; they should, since they were created by the Jim Henson creature shop and were the predecessors of all the current systems used for their full-bodied animatronics.
What an Idiot: The two footsoldiers that tried to drown Donatello in a fishtank. The look on his face while underwater screamed "How stupid are these guys."