During the finger amputation the doctor proceeds without using one iota of sterile technique, most notably no mask or gloves, let alone Betadine antiseptic which would have left the hand and forearm bright red.
When Carrie and Robert are interrogating the suspect, Carrie grabs scissors to threaten him. In the close-up for the hands, Carrie's left hand is completely visible with all her fingers. She had two of them removed previously.
When Carrie gets the dice, she switches instantaneously from holding the box in front of her (shots from behind Kate Beckinsale) to having her arms crossed with the box in her palm (shots from in front of Beckinsale) several times.
When Carrie struggles to button up her cardigan, Robert Pryce assists her in buttoning it all the way up. Directly afterward, Carrie rushes into the hallway and we see that her cardigan is now unbuttoned to mid-torso.
Kate's pistol changes in the middle of scenes. Early on, she opens a file drawer and we see a small Glock pistol in a shoulder holster. She has a flashback and then retrieves a different, full size pistol in a belt holster from the drawer. Later in a outdoor battle, she pulls the large pistol from a shoulder holster. When she drops the pistol we see it is again the small Glock.
As the doctor makes the quote about Scotch, he puts the bottle against his shoulder with the label facing against it. When the camera then shows a front view of him, the label is now facing away from his shoulder.
When the movie begins we see Kate Beckinsale's character walking outside. The camera follows her as she walks indoors and down a hallway. In one of these scenes we see her talking off her winter gloves in the hallway, but in the next scene when she enters her room, we see that she still has her gloves on.
The station leader Sam points to Haworth Mesa on the map as being in the Australian Antarctic Territory of east Antarctica. Haworth Mesa, where Weiss was found, is in the Trans-Antartic Mountains of west Antarctica approximately 1300 km south-east of McMurdo. Weiss' two companions then turn up at Vostok in east Antarctic, 1600 km away.
In the opening sequence, the date is shown to be 1957; however, the Soviet transport aircraft is an Antonov An-12, which first flew on December 16, 1957. It did not enter Soviet Air Force service until 1959. Also, a character stated that (to paraphrase), "They haven't flown these for 50 years", which is absurd, since the An-12 is still in service with many military forces and civilian airlines around the world.
When the Marshall, the guy from the UN and the pilot enter the buried Soviet plane, they find vodka in perfect condition. However, graphics have shown the outside temperatures to be in the -60c to -65c range. Vodka freezes at around -27c to -40c depending on proof. The vodka would have frozen at some point before being covered by the insulating snow and the bottles would have broken.
The pilot follows the two characters down the hole to the crashed plane. In a search and rescue operation, one person would stay above ground to make sure that everyone can get back for just such emergencies as the hole collapsing.
Antarctic ice, specifically plateau ice, does not make drinks "boil" or effervesce. While ice has tiny bubbles trapped and compressed under pressure their release causes nothing more than a crackle and pop. This is more distinct in ice from lower depths due to greater compression, and is almost non-existent in ice near the surface.
The UN has no law enforcement coverage of the Antarctic, and has no authority to commandeer aircraft or any other facilities. The activities of personnel in Antarctica are governed by the laws of their home nation.
Antarctica is the driest continent with very little snow precipitation. It would take hundreds if not thousands of years to bury the Russian aircraft to any depth. A Russian aircraft crashed on the plateau near Mawson station in 1968 is still sitting on top of the ice.
Update: The plain was on fire, what makes ice melt, that could explain, why the plain is under the ice.
In the scenes with the helicopter and the C-130, part of the tail number is visible and starts with the letter C, which denotes Canadian aircraft; as this was supposedly a US expedition the aircraft tail numbers should start with an N.
The Hercules tail number, C-GHPW, is registered to a company in Yellowknife Canada.
For low altitude flight the cabin would not be pressurized. Although the bullets from the opening gunfight damage the plane's hull, this would not result in a catastrophic depressurization. When a round breaks the cockpit window, the result is not depressurization, but a shattering of the window and the wind caused by a speed of 100+ knots. Air and snow are moving into the cockpit, not out as would be the case under a sudden depressurization.
When Kate and the team went out searching for the lost plane, she's using a Garmin 60 series GPS. On the screen it shows the first page, which is the satellite signal page which also shows the longitude and latitude. By searching the coordinates that are displayed (50 deg 45'55.3N, 98 deg 22'61.2W) you find that they are in fact on Lake Manitoba, Canada; not the South Pole. (Lake Manitoba is a known film location.)
In the finale, the wind is strong enough to bodily hurl characters off into the distance, provided they are not attached to a safety line. As soon as they are attached to the safety line, the wind has comparatively little effect. The characters have no difficulty standing in or walking against the previously irresistible force of the gale.
The goof item below may give away important plot points.
Immediately after Carrie's two fingers are amputated, we see her push herself up from her cot directly on the area of the missing fingers, putting pressure right on the amputation site without even wincing. She also manages to push the cart into the door with two hands, probably not possible immediately after amputation.