Factual error: Science Officer Tilly calls the planet Trill a "very big" planet. It has an area of 500 million square kilometers. But that is not so large a planet. The Earth has an area of 510,100,000 square kilometers. So Trill is slightly smaller than Earth. Not really a "very big" planet.
Plot hole: Michael uses herself as bait to trap her future self, putting her own life in jeopardy with the reasoning that her future self will come back to save her. All well and good, except they have a backup plan with the doctor to resuscitate her if needed, meaning her life isn't really at risk, or nowhere near as much as might be implied. And her future self would undoubtedly know that, having lived through it in the past, so not swoop in to save her. Or even if she did come, would also know it was a trap.
Suggested correction: ***SPOILER ALERT*** But, as it turns out, The Red Angel that comes to save her is NOT Michael, but her mother, who would not necessarily have known about the backup plan.
***SPOILER ALERT*** That it was her mother doesn't stop it being a plot hole since they thought The Red Angel was future Michael, and future Michael would know that present Michael wasn't really in danger so they weren't presenting a situation, _according to what they believed_, that required future Michael to act. It being the mother was a plot twist that created a motivation to act that the present people had no reason to think would exist. Basically, unless they presume a split timeline (i.e. this present is a different past than The Red Angel lived through), making a trap for future Michael that present Michael is involved in makes no logical sense.
Alternatively, Michael would have to come back, KNOWING it was a trap, to prevent the timeline unravelling.
The point of the exercise is they were setting a trap. If it didn't work, then Michael wouldn't have to come back to "prevent the timeline unravelling (sic)", even if that were a thing - it presupposes a fixed, unalterable timeline, which goes against their attempt to send the data to the future to protect it, and thereby alter the future. Even with an unalterable timeline, it would only work if future Michael had chosen to allow herself to be trapped, but in that case why wouldn't future Michael just voluntarily come back to help? Since her being trapped wasn't a certainty, there was no reason to think she would be given that the current Michael, and therefore also future Michael, knows a trap has been set, but one that doesn't actually threaten current Michael. The whole premise of the trap, under their assumption that The Red Angel was future Michael, is completely flawed and made no logical sense.
The fact that The Red Angel was in the future, and that they had a backup plan meant that The Red Angel never should have come back in time, ever. Because the backup plan would be the recorded history, thus, she never would have died. Thus, nothing to save. Face it, everything in Discovery is a plot hole.
The War Without, the War Within - S1-E14
Other mistake: In dialogue, Starbase 1 is described as "100 AU from Earth," which is over three times the orbital distance of Neptune, but still within the Solar System. But the onscreen graphic shows the starbase next to a planet that can be visually recognized as Earth; portions of North America, including the Florida peninsula and the Gulf of St. Lawrence, are visible.
Such Sweet Sorrow: Part 2 - S2-E14
Character mistake: When Tyler arrives in the Klingon ship, Captain Pike says "calvary arrived just in time." The word is "cavalry." The subtitles show "cavalry."
Through the Valley of Shadows - S2-E12
Continuity mistake: As Gant is trying to inject Burnham's eye, in wider shots the syringe is noticeably further away from her eyeball than it is in close-up shots.
Continuity mistake: Michael invites Spock to play chess. She presents a board with three levels. Later, when the game is underway, the board has seven levels by the addition of four smaller boards. (00:17:50 - 00:21:25)
