The Adventures of Super Mario Bros. 3 contains examples of:
Absentee Actor: Between four heroes, one Big Bad and seven Koopa Kids, there were bound to be some absences. "Life's Ruff" is a special case. Most of the regulars are absent, but so is Mario himself—making for the only episode of the three shows that he didn't appear in. (Incidentally, this means Luigi is the only character to appear in every episode of every show.)
Arson, Murder, and Jaywalking: In "Crimes R Us," the Koopa Kids steal the following from the Mushroom people: 522 wallets, 391 purses, 912 gold coins and one yo-yo.
As Long as It Sounds Foreign: Parodied in "The Venice Menace" in which the Italian language is made up of nothing but names of famous Italians and Italian-Americans. Hilariously, one of the names uttered is "Lou Albano", AKA Mario's old actor/voice actor (he and Danny Wells were replaced by the time of this cartoon).
Badbutt: Bully Koopa, who's portrayed as a kid-friendly gangster type.
Bag of Kidnapping: In a rare villain example, the Koopalings are caught by the Mario Bros. robbing the Mushroom Kingdom Treasury in "Crimes R Us." King Koopa then opens a dollar-sign bag and is shocked to discover the Koopalings gagged with their hands tied behind their backs.
Extra Hindsight Hilarity points for the fact that when Milli Vanilli point out they can't be forced to do a song without their backup singers. Taken further in that when Kootie Pie gives up on having them as her captives, she laments that their music was too beautiful to last!
Co-Dragons: Kooky and/or Cheatsy are the ones most likely to be on hand in aiding their dad.
Comically Missing the Point: In "Do the Koopa", Koopa is using the magic music box to force everyone to dance, including his traitorous kids.
Deus ex Machina: The wands were used like this a couple times, but this was averted in "Recycled Koopa," where magic didn't fix everything.
Disability Immunity: Bowser has a cold in "Do the Koopa", rendering him hard of hearing. This means he can't hear the Doom Dancer Music Box's music, and therefore isn't affected by it.
Divide and Conquer: Said word-for-word in "True Colors" as the Koopas plan to turn the Mushroom people against each other.
Dub Name Change: The writers made up their own names for the Koopalings, called the Koopa Kids in the actual show, before the English version of The Manual was final. That, and some of the references Nintendo ended up using would likely go over most kids' heads anyway.
Easily Conquered World: Earth itself. Several episodes feature King Koopa taking over a country or a city with minimum effort. One even has each of the Koopalings taking over the continents, with nothing more than their magic wands.
Episode Title Card: Done in the style of the title screen for Super Mario Bros. 3 (a curtain raising), naturally (except "Kootie Pie Rocks").
Even Evil Has Standards: At the end of "Princess Toadstool for President." Koopa not only loses the election, but only gets one vote. His kids and his own army didn't even vote for him.
Cheatsy: We couldn't help it, King Dad. We had to vote for her. She's so nice and you're so... so... (Koopa growls angrily) Exactly, Dad.
Expressive Mask: The eyes on the frog suit open and close at the same time as the wearer.
Fanservice: Princess Toadstool wears a swimsuit in more than one episode. Sure, it started as a Beach Episode, but at one point, she gets captured wearing it and spends the whole rest of the episode clad as such.
There's also her Totally Radical '80s outfits in the episode "Kootie Pie Rocks".
Fantastic Racism: Exploited by the Koopas in "True Colors", who paint the townsfolk red and blue and spark riots between them.
Mario: You're so stubborn now that you're red, Luigi! Luigi: Yeah? Well, you're a blue know-it-all!
Foot Focus: A barefoot Princess Toadstool in "Reign Storm".
Heel-Face Turn: During the episode "Crimes R Us," King Bowser Koopa abducted Crimewave Clyde out of prison to train his kids how to become hardcore criminals. They became too much for Clyde and after the Koopalings double crossed him with the Mushroom Kingdom Treasury, he switched sides with the Marios and gave Bowser and his kids a lesson in payback. Though Clyde did return to prison, he was more happy to 12,423-year prison sentence with no parole than to spend another day with the Koopas.
King Windbag of Ice Land also had one in "Life's Ruff." Princess Toadstool accused him of being a bully and sent Luigi to his castle. He witnessed the king mistreating his servant and threatening him with his wand. He and Luigi were turned into dogs by Hip and Hop and were eventually sent to the pound by a dogcatcher. King Windbag was cajoled by Luigi to work with the other trapped dogs and together they escape. After turning himself and Luigi back to humans, he arrived back at Ice Land with a warm heart and prepared a feast for his servant.
Idiot Ball: Princess Toadstool picks it up in "Never Koop A Koopa." When King Koopa pretends to give up and hands her the key to Castle Koopa, she decides that she'll turn the castle into a retirement home for elderly mushrooms and runs off to check it out.
Leitmotif: Since the show re-arranges most of the music from the game, the tunes are indicative of the character or type of environment that is currently on-screen. Examples: Water Land's theme playing any time there's a swimming scene; Dark Land's theme playing whenever King Koopa or his fortress is on-screen; the airship theme whenever Koopa's airship appears. And appropriately, most episodes ended with the credits theme from the game.
Lolicon: Subverted in "The Beauty of Kootie": Mario uses Kootie's wand to transform Luigi into "Casanova Koopa" to distract Kootie while Mario, Toadstool, the King of Desert Land, and Toad escape. She kisses him on the lips passionately, which disgusts Luigi himself while in disguise.
MacGuffin: The wand in "A Toadally Magical Adventure."
Magic Music: The Doom Dancer Music Box made anyone who could hear it dance uncontrollably. This became a plot point as Koopa, who had a cold, couldn't hear it and thus wasn't affected when Bully tried it on him.
Make Up Is Evil: In "Reign Storm" the Koopas create an evil robot version of Princess Toadstool. Who wears really dark lipstick.
Mama Bear: In "Mind Your Mummy Mommy, Mario," Queen Mushroomkhamen becomes enraged when her son's sarcophagus is stolen.
Meaningful Name: All of the Koopa Kid's names for this show. Some (such as Bully, Big Mouth and Cheatsy) are more blatant.
Mermaid Problem: In one episode Mario was rescued by a mermaid named Holly Mackerel, who fell in love with him. Complicating matters was the fact that she was a reverse mermaid, and that she saw him in his Frog Suit, which made her think he was a frog.
Mind Control Music: The Doom Dancer Music Box makes anyone who could hear it dance uncontrollably. It can also be used to march others away, and if cranked slowly, makes them slow down and stop moving.
Near Villain Victory: Koopa came THIS close to winning in "Never Koop A Koopa." He had everyone captured and at the mercy of Bob-Ombs...and lost cause Cheatsy decided to cheat him.
He came as close in "Crimes R Us." The entire Mushroom Kingdom Treasury could have been Koopa's for the taking had it not been for Crimewave Clyde foiling the Koopa Kids' plan to destroy him and his subsequent Heel-Face Turn.
Off Model: As with the Super Show, Mario and Luigi still appear off model with their overalls and shirts having the colors swapped, despite that the appearance they have today was used in the game this cartoon was based on.
It seems they were intentionally mocking this trope, since when Koopa sends Kooky to Hawaii to make sure the Princess stays there, all he gives him to fit in with the humans is a lei to wear around his neck and it works.
King Koopa and Kooky dress up in plumber's clothes in "Toddler Terrors of Time Travel" and proceed to clog an apartment's drainpipes.
This doesn't work for the good guys. In one episode the Mario Bros. try to disguise themselves as sledge brothers to stow away on the Doomship. Luigi is worried that it won't work, Mario tries to assure him that he barely recognizes him. But once they get aboard, an actual sledge brother deadpans "Puttin' on a little weight, aren'tcha Mario?"
Portrait Painting Peephole: Demonstrated in "Never Koop a Koopa" when Koopa watches Princess badmouth him through the eyes of his own portrait.
Shown Their Work: This show was the closest adaptation of the games, even featuring P-Wings and the Kuribo's Shoe in an episode, as well as the ruler of Desert Land. Ice Land's ruler is also featured but is given a different appearance from the game.
The music box in "Do the Koopa" may reference a similar item from the game.
Slow Transformation: Seen in "Recycled Koopa" with the humans' transformation into Koopa zombies.
Space Whale Aesop: "Recycled Koopa" is supposed to be about recycling, but that aspect only shows up at the very end after Koopa's trash has turned the inhabitants of Brooklyn into zombies.
A Storm Is Coming: In "Oh, Brother!" Mario and Luigi's argument hits its peak as it starts to rain.
Super Power Meltdown: Toad and Princess Toadstool use a filled-to-the-brim chest of power-ups that Toad well-stocked in "Super Koopa" to keep Mario and Luigi powered up each time King Koopa de-powers them, though they sometimes cause Koopa to be de-powered. Both sides use their power sources (a pendant for King Koopa, the chest for Mario and Luigi) to keep the fight going. The constant stress of this, however, causes King Koopa's pendant to short-circuit, and we see him turn into many powered-up King-Koopa versionsbefore he finally explodes into his normal de-powered self, the pendant useless as he's sent through a manhole into the Mushroom Kingdom again.
Title Drop: "Mush-Rumors" featured Mario saying the episode title at one point.
And also in "True Colors".
Took a Level in Badass: Princess Toadstool took a more active role in the adventures and was less prone to kidnapping than in the Super Show. She and Toad were even able to use power-ups in the same way as the Mario Bros.
Big Mouth: It's curtains for you creeps! You're history! You've had it! You're done for! It's all over! We win! Time's up! Your clock is cleaned! The sun has set! The fat lady has sung! The cheese has molded! The Super Mario Bros. are no more!
Water Is Air: In "The Ugly Mermaid", King Koopa tries to make a kingdom of fish surrender to him by flooding their air-filled dome with water, to which they react negatively for some reason.
The merpeople even have to wear water-filled globes over their heads so they can breathe while in the city, Seriously, what?