At the end of the film, Danny is loaded into the ambulance correctly (head first). As it drives off, we see him and his wife in the ambulance and his body is now in the opposite direction, as if loaded feet first.
The taped segment of the conversation between Danny and Nate before being interrupted is different from the actual scene at the beginning of the film.
Shooting a computer monitor doesn't destroy any data that lies on the computer. Furthermore, after the shootout in which Nibaum gets killed, Danny recalls computer being destroyed and implies that the evidence is also gone. This is also most likely false, as there are only few bullet holes visible on the computer, which unlikely destroyed the hard drive(s).
When Danny and his partner first talk in the car outside the party, there is no rear view mirror in the car.
At the end of the fight between Danny and the corrupt cops in Internal Affairs, Danny looks at them when he hears one of them has run out of bullets. When he looks at them, the door is pulled wide open, but the next shot shows them just pulling it open.
The sharpshooter puts a laser dot on Roman's head and the officers in the helicopter light Roman up with about eight lasers while he's standing in the window. Police sharpshooters don't use laser sights (i.e. red dot on target). They're good enough to not need them and don't want to tip off the suspect that he's in their sights. Any cops who do have laser sights (which is unlikely, but conceivable) don't put them on the target and keep them there. The sights are for rapidly getting the gun on target - not replacing proper sight picture.
Roman asks that the electricity be switched back on. When the order is given to restore the supply, the audience can hear clearly the sound of the generators starting up. Something that would never be heard in the office block.
Police negotiators and entry teams conduct their activities independent of one another so that the negotiator doesn't inadvertently let the hostage taker know what the entry team is doing.
When Roman fires the (first) shot into the ceiling inside the building where he is holding 4 hostages, the crowd and news crews outside react to the sound - as do the police who are monitoring sound inside - however, the sound of a handgun shot inside a building and several floors up would not be heard down on the street below (nor would the muzzle flash be visible).
Obviously the last shot in the ambulance is to show the scene and the patient leaving the place, but a patient inside an ambulance rides with the head to the front.
During one of the early negotiating scenes, the clock in the negotiating room shows early afternoon when it is clearly night time.
When Palermo is told to take the shot, a closeup on his eye blinking through the scope is a stand in. This is evident by the brim and strap of a helmet and facial wrinkles that Palermo doesn't have. He has smoother skin and wears a backwards baseball cap.
The fire sprinklers only activated when the HBT and FBI entered the office room. They should have started once Roman started the fire.
When Danny is shouting from the broken window the helicopter's noise would make his words hard to hear.
When Roman and Sabian is browsing Niebaum's computer at his house the Address bar doesn't match the content of the other windows. The address bar says C:\niebaum\office files\ongoing investigations\*.* while the All Folders sidebar shows that C:\Office files\Accounts is selected. C:\ is labeled Niebaum, there is no C:\niebaum\ folder.
While browsing A:\Ongoing Investigations the explorer window says (A:) - Office files - Ongoing Investigations which doesn't match the contents of the All Folders sidebar.
Furthermore, although A: is full of documents they're marked with the windows icon, proof that no program has been assigned for opening or editing the documents as they were most likely done in haste.
The waveform being shown of the audio file being played is way off in comparison to the actual audio itself.
Although the movie is set in Illinois, two of the volumes in Niebaum's office are from Ohio Jurisprudence.
The ambulance at the end of the movie is an AMR unit. AMR does not operate in the City of Chicago (nor Cook County). That should have been either a Chicago Fire Department or Superior Ambulance.
The building where Danny holes up is at Wacker and Clark; police cars seen, ostensibly "to the scene," are actually speeding south on Wabash at Van Buren - a point already south of the building.
When Sabian is relieved of his command, he runs into the building to talk to Danny. The FBI and the police know this, yet nobody stops him on the stairwell, as one would expect.
Before Danny takes over the building, he tells Karen to wait in the car and that he'll be right back. Yet she never finds out about his takeover until much later at nighttime. She most certainly would have learned immediately through word of mouth about her husband being a hostage taker not very long after it happened.
Sabian, an expert police negotiator, brought in an actor to act as Nathan's informant. A proper police negotiator would never use an actor or lie to the hostage taker which would result in an adverse scenario. The negotiator should just negotiate and connect with the hostage taker, understanding him or her, never lie.
It was totally unwise for the whole Team to include Farley, a rookie police negotiator. In the first situation, the audience learns he cannot get the personal details of the dog Omar owns. Secondly, he was nervous and said 'no' which was in the police manual nearly causing Roman to shoot a hostage. Police negotiators go through detailed training before deployed.