One of the major ones is Dr_Death, the operator of the pound.
Quoth a farmer from the Altador Plot: "WINDMILLS DO NOT WORK THAT WAY!" There's even a book given out from the plot titled "The Way Windmills Work" with the following description: Windmills do, in fact, work that way.
In one of the quotes seen just as you enter a battle in the Neo Cola Centre sector of the Battledome is, "Would the owner of a white space ship, designation MCP1138, please pick up the white courtesy communicator. You\'re double parked."
While visiting Coltzan's Shrine, you will sometimes see your Neopet say, "And to think, Malkus Vile would have gotten away with it if it weren't for us pesky Neopets!" This is a pop culture reference to Scooby-Doo.
In the "Spooky Food Eating Contest" plot, the monster hunters Kell and Corbin are expys of Sam and Dean of Supernatural, respectively.
The Perilous Catacombs step has a few:
One of the rooms you can find if you choose the right option when opening a door is a treasure room. The text for it includes, "You're all for taking a swim in the gold coins littering the floor, but Corbin and Kell are remarkably uninterested."
One of the curses you can get is the Curse of Barrel Roll, which causes the map of the floor you're on to spin while the text "BARREL ROLL!!!" is displayed in the middle.
Occasionally, one of the options for opening a door is to play the banjo. If you choose it and are successful, you "launch into 'The Pant Devil Went Down to Altador' and play your banjo fiercely." This is a reference to the old song, "The Devil Went Down to Georgia".
"We need to go deeper.", which 'you' say when you're ready to go down the next floor, is a Shout-Out to Inception.
Corbin cries out Ackbar's "It's a trap!" when barge in on the doppelgangers when you hear banjo music.