(at around 47 mins) When Professor X is telling Beast about how he met Raven, the whiskey bottle on the table rotates between shots.
(at around 12 mins) The damaged shuttle is spinning because a thruster is firing. When this is destroyed, it slows down to a halt almost instantly. In the vacuum of space, this would not happen. It would remain spinning fast, while decelerating slowly due to the opposing thrusters firing. Conservation of angular momentum in action.
The film is set in 1992, 30 years after the events of X-Men: First Class (2011), yet the characters have barely aged during the intervening years. A few mutants have anti-aging abilities, but not all of them. This movie is set in an alternate timeline, so all continuity errors are justified.
The opening scene of the movie is set in 1975, but Warren Zevon's "Werewolves of London" is heard on the radio, which wasn't released until 1978. However, due to the time traveling events that Wolverine did in X-Men: Days of Future Past (2014) which altered the time line. It caused a ripple effect in which Zevon might have recorded the song earlier, possibly from seeing Beast in his blue form.
(at around 5 mins) When Jean is reading Charles' mind in New York, in the flashback, Charles can be seen driving an eight-year-old Jean to the school, but it was established that he was in a wheelchair then and had no control over his legs, and therefore couldn't drive. But X-Men: Days of Future Past (2014) revealed that he sometimes uses a drug to regain control of his legs (at the temporary expense of his powers), so it's possible he took minimal enough of a dose to be able to drive the car.
Jean's and Nightcrawler's mutant abilities do not allow them to survive space. The X-men save astronauts who are wearing protective suits while wearing leather jackets.
The basketball court opens when the X-Men's aircraft is about to launch. The goals move rigidly. When fully opened, the goals have disappeared with nowhere to have gone.
(at around 1h 40 mins) At the end of the movie Charles is seen wearing a modern (2007+) Rolex Milgauss.
When Jean sits in a bar posing as an old man, several bottles of whiskey are shown. One of those, the GlenLivet Founder's Reserve, was not available until 2014.