There are a few references to An An American Werewolf in London (1981). A movie marquee reads "See You Next Wednesday" (which is also a running gag for John Landis' movies), David Naughton plays "Ambassador Kessler", and there is a tavern in London named "The Slaughtered Lamb". "David Kessler" was played by Naughton in An American Werewolf in London (1981).
(at around 6 mins) When Fin and Nova are exploring Stonehenge, props from Season of the Witch (1982) are seen (the skull mask and a Silver Shamrock Novelties logo badge). In Halloween III: Season of the Witch (1982), Stonehenge was used as power for the villain of the film's plot. In this movie, the two items are seen very briefly laying on the ground within the first several minutes of the film, when the two characters, Fin and Nova, are deep within the caverns of Stonehenge. A keen eye is needed to catch this, if not, this very minuscule detail will be missed.
(at around 3 mins) In the beginning of the movie, Fin rappels down below Stonehenge, stopping in a prone position just above the ground, and says, "Mission accomplished. Eat your heart out, Cruise." This is a reference to the scene in the Mission: Impossible (1996) movie where Tom Cruise's character ends up prone, hovering dangerously close to the temperature sensitive floor in a secret vault.
(at around 7 mins) When Fin and Nova are under Stonehenge looking for the Druid artifact, they have to jump through a chomping giant shark jaw. Just before they jump, Fin says, "Why does it always have to be sharks?" This is a reference to Raiders of the Lost Ark (1981) and Indy's line asking, "Why did it have to be snakes?"
(at around 27 mins) The Secretary General, played by Nichelle Nichols (Uhura, Star Trek: The Original Series (1966)), tells Fin that he has gone "where no man has gone before" - this is an iconic Star Trek line spoken by an iconic Star Trek actress.
(at around 45 mins) In a brief comedic exchange between Electra and Orion (Olivia Newton-John), the phrase, "tell me about it stud" was cut short by Orion saying, "no, don't!" The phrase was used by Newton-John's character Sandy to Danny (John Travolta) in the movie Grease (1978).
(at around 7 mins) The light-sensitive trap in the Shark God's temple is an homage to Raiders of the Lost Ark (1981). In the trap, Nova and Fin find Indiana Jones' fedora and whip, implying that the skeleton wearing the items is Indiana Jones himself.
(at around 15 mins) The British Prime Minister (played by Chris Kattan) retrieves a silver briefcase containing the trigger that can launch the UK's secret Sharknado Defense system, referred to as Skyfall. Skyfall (2012) is the name of the 23rd James Bond film.
(at around 52 mins) In Rio, Fin and April are taken to meet someone named Vega. Just before they enter the building for the meeting, April says to Fin, "I have a bad feeling about this." This is a famous line from the Stars Wars franchise, and is uttered at least once in every Stars Wars movie.
When Dr. Angel (Geraldo Rivera) opens his safe, he finds it empty and says that he'd swear there be a bottle of liquor inside. This is a jab at Rivera's own documentary The Mystery of Al Capone's Vaults (1986), in which the vault of gangster Al Capone was located and opened, in the hope of finding his secret treasure or even a corpse. The show was so hyped in advance that it garnered one of the highest Nielsen ratings in television history, but it ended in disappointment because all that was inside the vault was dirt, an empty bottle of liquor and a roll of quarters.
(at around 26 mins) Following the devastating Sharknado attacks in America, Fin addresses an emergency meeting of NATO. He says "Back home we're trying to make America great again". Sounds like a slogan from the 2016 US Presidential campaign.
The trivia items below may give away important plot points.
(at around 41 mins) When the Sharknado drops Fin, April and Nova in Australia, they drop her off at an ultra secret organization made up of women. When asks if there is anything on the internet, the other girl says "check everything, except stay away from Cyberdyne!" That is the company from the Terminator film franchise that tries to destroy humanity.
(at around 39 mins) In Sydney, Australia, April gets attacked by a shark and is broken into pieces (specifically, her legs become detached from her torso). In the next scene, April is being carried by Fin on his back in his back pack, exactly the same way Chewbacca carried C3PO in Star Wars: Episode V - The Empire Strikes Back (1980) after he was broken into pieces by a blaster on Bespin in Cloud City.
(at around 5 mins) April and little Gil enter the laboratory of weapons master Llewellyn, accompanied by a music theme reminiscent of many Bond movies. In the early Bond films Q was played by Desmond Llewelyn.
(at around 31 mins) When the Zeppelin, known as the Hindenburg, hits the Sharknado, it's destroyed in the same manner as the original Hindenburg in Lakehurst, New Jersey. Even with the someone saying the famous line, "Oh the humanity!"
(at around 16 mins) Poison front man Bret Michaels has a brief cameo where he stays off a shark with a Sharkslayer guitar, on the front bumper of a double-decker tour bus. This is most likely a reference to the flame-throwing guitarist on the front of the vehicle in Mad Max: Fury Road (2015).
(at around 1h 28 mins) The last line in the movie is, "Where we're going, we don't need a license." This line mimics the last line spoken in Back to the Future (1985): "Where we're going, we don't need roads."
(at around 9 mins) When Fin and Nova are under Stonehenge, they find a Druid artifact that they believe can help them defeat the Sharknados. Nova reaches out to take the artifact, but Fin stops her, saying, "No, we need a counter balance." He then proceeds to switch the artifact with Nova's bag of Megalodon teeth, hoping the weight is the same as the artifact and avoiding a possible weight-sensitive booby trap. Indiana Jones does the same thing (except with a bag of sand) during the opening sequence of Raiders of the Lost Ark (1981). Fin's attempt to use the counter balance fails (as did Indy's) and they are forced to run for their lives. As they flee, you can see a large golden shark head statue rolling behind them, just like Indy ran from the giant rolling boulder as he fled following his failed attempt. When they finally get to their escape helicopter, the pilot is called Jock, just like Jock the pilot who helps Indiana Jones escape the Hovitos in Raiders of the Lost Ark.
(at around 1h 28 mins) At the end Fin and Gil take off in a vehicle whose wheels go horizontal as does the car in Back to the Future (1985). The final "TO BE CONTINUED" uses the same graphics, font and graphics as Back to the Future with the addition of a shark's head.