When the men start hauling their boats over the ice, we see the name "J. Caird" clearly painted on one of them. A few minutes later Shackleton announces he has decided to name the boats and unveils the painted names for the first time.
Shackleton is shown wearing the uniform of a Lieutenant in the Royal Naval Reserve, when in fact he only served as a Sub-Lieutenant and had resigned his commission in 1904 and was therefore not even eligible to wear a uniform.
When Shackleton lifts up the binoculars to watch the Endurance being crushed by ice, a cameraman operating a camera is reflected in the lenses of the binoculars.