Continuity mistake: When the girls are leaving the Statue of Liberty the first outside view of the ferry is white and says "N.Y. Waterways." When they show the second outside view it is a red ferry with no title.
Other mistake: After the explosion at the top of the Empire State Building, when Fin's soaring through the air he grabs hold of the chainsaw and powers it up to use on the sharks, but in one shot the cutting chain has been removed from the chainsaw's guide bar. Then the cutting chain is back, but it's not moving even though we hear it working.
Visible crew/equipment: While Fin's on the Empire State Building's observation deck Skye heads down to make a freon connection, and right after the closeup of the shark in the sky heading for the building, when it cuts to Fin we can see the reflections of the camera, two crew members, and a red step ladder with a yellow top (possibly a Werner ladder), all reflected on the glass door. (01:16:50)
Continuity mistake: While on the 7 train, when Fin looks out the last car's window and sees the flooding and oncoming sharks, as he shouts, "We're in trouble!" in the closeup we see the door has certain stickers and markings and there's an advertising poster on the wall, but when it cuts to the wide shot it's all changed.
Revealing mistake: After Fin, Martin, and the rest get what they need in the Times Square area, they get in the cab and drive off, and just as we hear on the news "Moments ago the water spout came ashore and devastated the UN," in this exterior shot we can see the taxi's back tire as it's driving down the street, but the tire's not actually moving. Presumably the taxi is on a flatbed being driven by crew while filming from this angle, hence the tire being stationary.
Continuity mistake: After April sneaks out of her hospital room, it cuts to the taxi with Fin, and the cab pulls over to the curb at the corner of Broadway and West 48, but in the closeups they're in an entirely different area - note all the buildings, then in the wide shot they're back on Broadway.
Revealing mistake: When the stewardess asks April to sign her copy of the book entitled "How to Survive a Sharknado and Other Unnatural Disasters..." it's a thin book, and in the closeup as April autographs the page we can read the text and it's actually one of the versions of the "Samsung Galaxy Tab...For Dummies" book, not April's book. Also notice when Kelly Ripa shows April's book to the audience, it's now at least 4x as thick.