Amanda mentions planning on going to the Vulcan High Command to look for Michael when she ran away. However, the High Command was dissolved in 2154, over a century earlier.
When Burnham and Amanda first meet with Pike in his office, the camera looks in from the adjacent corridor through an open doorway. When the camera angle changes to a view from within the office, the door is suddenly closed. It remains closed throughout the rest of the scene.
The force field bubble around the fungal organism in engineering forms before Saru finishes giving the order.
When Burnham and Amanda are reviewing Spocks' medical files the monitor behind Burnham, as she is about to compile a search record of the problem associated with Spock, shows the Sphere which they, Discovery did not encounter until the following episode. Clearly a mistake.
Ash Tyler says he's heard of the black badge used by Section 31 but never seen one. There were black-badged officers serving aboard Discovery in the first season.
During the marathon, Tilly is stopped by her conversation with the May apparition for the better part of a minute. When she rejoins the race, the other runners have barely moved from where they were when her conversation with May started. Her hallucinatory conversations are shown to play out in real-time, so the other runners should have been much further ahead, probably finished with the race by the time she started running again.
Stametts says Tilly is harboring a eukaryotic organism, which is taken to mean a fungus. This is not what "eukaryotic" means -- eukaryotic organisms includes all fungi, plants, and animals, and presumably many of the extraterrestrial species on Star Trek.