When Cameron waits for Ba'al on the ship he has two M1 Thompson Submachineguns. The one he gives to the crew member, the M1 Thompson, was not developed until after 1939 during WWII. They should have been 1921 or 1928 Thompsons.
When Jackson gets out of the cab, his leg appears to have been amputated above the knee. Later, when his prosthetic leg is shown, it seems to be designed for someone whose leg has been amputated below the knee. Further confusing matters, when he walks into the bookstore both legs bend at the knee, despite the fact that his left leg seems to have neither a knee nor an artificial knee joint.
Dr. Jackson's prosthetic leg seems to have a clear plastic "fitting" socket. These sockets are used only to ensure that the final socket (made from carbon graphite, or more rarely fiberglass) fit properly. Fitting sockets are not durable enough for daily use.
While Carter and Mitchell are walking through the Arctic, a helicopter can be seen flying over one of the hills in the background as the shot fades to the next sequence. However, since the US military is conducting a training exercise in the area at that time (as Jack confirms later), there is no error in having a helicopter flying around.
The F15s SG1 take to Antarctica are described as having "minimum armaments and extra fuel tanks" and that is how they are configured on the runway In the refueling scene though they have missiles loaded and not fuel pods. The other in-flight shots have a variety of different configurations probably because stock footage was used.
The goof item below may give away important plot points.
Incorrectly regarded as goofs
Near the end of the film, there is another scene in the cargo hold of the ship that shows the Stargate being activated in the past. The boxes in front of the crate containing the gate are arranged in a different manner when compared to the first scenes in the ship's hold. However, the boxes were probably moved by Mitchell to create a barricade for himself and to clear up the place around the gate to get clear shots. There was no one to move the boxes the first time around.
After activation, the wooden crate that the stargate was stored in has circular holes on both sides. The vortex that would create these exits only on one side of the ring. Or at least it usually looks like it. In the first motion picture, however, the vortex wobbles back and forth before stabilizing. In this movie the gate is shown head-on, so it's impossible to say what's going on behind it. But even if the vortex exists only in front, it may cause air disturbances strong enough to destroy the crate wall right behind it. There is no specific claim in the series that standing right behind a gate being activated is perfectly safe.