Near the end of the movie when Madeleine White visits Mr. Case at the barber, Mr. Case says "Yes, it was sixty years ago, I was young and ambitious", you can very clearly see his lips mouthing "fifty" instead of "sixty", though the audio says "sixty".
As Dalton stands inside the bank vault staring at the cash, the scene jumps from a wide, distant shot, to medium, to close-in, just outside the cage. Another robber stands just outside and between jumps his gun position shifts from relaxed to raised to relaxed again.
When Madeline and Case are talking next to the river, Case is standing with his right shoulder towards Madeline. In the next shot, he is turned dramatically so his left shoulder is towards her, and then it is back the other way.
Miss White's assistant's hair is unkempt and has hair forming two loops sticking up toward the back of his head when he tells of Mr. Case's phone call. In subsequent shots his hair appears combed with no hair sticking up.
When the ring case in the safety deposit box is opened, the Cartier logo is smaller, with "London" and "Paris" visible under the logo. Later, when Detective Case shows the ring in the box to the mayor, the logo is a different Cartier logo.
When the NYPD ESU is initially setting up around the bank at 20 Exchange Place, Manhattan, FDNY Engine 332 arrives on the scene. FDNY Engine Company 332 is based in Brooklyn, on Bradford St., not Manhattan.
In the last scene, Frazier empties his pockets to end the day. He takes his magazine out of his gun to disarm it, but we see there is no ammo in the magazine. A detective would never carry an unloaded firearm.
At the end of the film, Dalton is leaving the "storage room". On the left side of the screen, one can see a set of doors that have glass over them, and a short distance back, outside windows with sunlight streaming in. The distance is too short to match up to the apparent length of the storage room as seen in the earlier parts of the film. There is also no indication that there could be any ceiling over that space.
The goof items below may give away important plot points.
The robbers sent their demands written on a desk drawer from the bank. Later on Detective Frazier discovers that the drawer was bugged but this time the writing and capitalization is different. For example "SemTex" changed to "SEMTEX".
When the bank robbers first walk into the bank they are all wearing painter outfits. Upon bringing all the hostages into the basement the robbery crew make them all take out their cell phones and keys, and remove their street clothes, which the police all find in the bank after the robbery. While the police interview each hostage to see if they knew who was who, they supposedly had to just match each hostage with their possessions. But Steve, Stevie, and Stevo-O undoubtedly put their own street clothes, cell phones and keys in the bags along with those of the hostages, to claim as theirs after slipping out of the bank with the hostages.
In the scene where the 'hostage' is dragged kicking and screaming by his ankles through the rooms of the bank, he pulls a pile of books over that are stacked next to the doorway. Later in the film when the police storm the building, the books are neatly stacked back at the side of the door. However, it is entirely possible that either one of the robbers or hostages stacked the books back up during the course of the film.
When Frazier is in the bank to negotiate, he says "What happened? Couldn't you open the vault?" As he says this, the vault's door in the background can clearly be seen and is wide open. This comment is deliberately said in front of the open safe, as Frazier is being ironic. Much of the dialogue throughout the film is ironic, especially Frazier's. He had already commented that he realized the goal of the thieves is to gain time, and this is his way of saying to Russell that he knows.
The police conclude as that no money was taken, "no crime was committed." However, armed robbery is itself a crime, even if nothing is taken. So is taking hostages and controlling the building illegally for several hours, as well as other subtle misuses of bank property. "No crime was committed" is an unrealistic description of what happened.
When the hostages are lying on the street outside the bank and are being put in PlastiCuffs (also called FlexiCuffs), you can see that several of them are not tightened, so they can just take their hands out of them if they wanted to.