The events of Despicable Me take place during the month of May in (presumably) 2012. The day of the recital, the theft of the moon, and Vector kidnapping the girls, is May 26, which falls on a Saturday.
When the Shrink Ray is tested in Asia, the elephant being used as a test subject is an African elephant.
At one part where the minions are holding Gru from falling onto Vector, one of them initially has a combed hair. But when the shark makes them fall down from the vent, he has a buzz-cut hair.
The name "Bob" the minion appears in the credits, but there's no clear clue of which minion the name is referred (another Bob appears not until Despicable Me 2).
Though never explicitly stated in the movie, Gru's first name is Felonius, displayed in his family tree (partially obscured in the movie, though his name in the fake credentials is Felonious). Subsequent supplementary materials such as Minion Mayhem indicated that this first name is real.
Super Silly Fun Land, the amusement park in the movie where the inner father of Gru is awakened, is most likely based on Pacific Park in Santa Monica, California. Both parks are built on a pier, feature one roller coaster, and has multiple game stands. Adding to the credibility of this resemblance is the fact that Illumination Entertainment is based in Santa Monica.
There are issues with time in Despicable Me. Such as:
When Gru enters his home in the beginning of the film, it is 3:30 PM, according to his DVR clock. Yet just a few minutes later, the sun has already set and the Moon is in full display when Gru opens the top hatch of his underground lab. Though, according to the screenplay of the film, it may implies that it takes hours to assemble all the minions.
As Gru walks up to the counter in the Coffee shop he is holding his freeze ray but in the next shot it is gone.
The minions take out many things for sale to support Gru, and there's a crown, but it's different from the one appears in the film Minions which is taken away by Gru.
In the end credit, John mimics Gru by saying "back to work", which is reused in a slogan on a poster of Despicable Me 2, written as "Back 2 Work".