Well, discounting Discovery since it's a streaming one, Nearly all of Enterprise. TOS with its canned episodes never really showed the consequences of individual episodes and things were sort of glossed over.
In "Where No Man Has Gone Before" Kirk's name was James R Kirk. After that, it was James T Kirk. Minor, but still a continuity break. Little things like that were plastered across all series.
I don't know why they changed his middle name, but that's incredibly minor, in the context of the franchise.
Nearly all of Enterprise was continuity breaking, even with the explanation of Temporal Cold War. Did all of that somehow get labelled top secret enough that afterward, nobody knew any of the events? The appearance of the cloaking device, which in canon was "Balance of Terror" suddenly appeared much earlier in Enterprise, yet was still a massive surprise to Star Fleet in Balance of Terror.
I don't really remember much of the prequel, TBH, but I find it hard to believe there's no explanation for that. Not saying it's not a valid point, just that I don't remember enough to counter it.
The appearance and nature of the Trill changed over time too. That breaks inner continuity. We won't even go into the appearance change of the Klingons in Discovery....since again, Discovery.
Appearances are aesthetics, and the changes could be due to better makeup techniques, or simply different makeup artists' or directors' visions of what the characters should look like. I don't know what you mean by the "nature" of the Trill changing.
What about the Q episode in TNG where the girl (Amanda Rogers) finds out her parents are Q?....then in Voyager, Q states that he wants to be the first Q to reproduce?
Didn't they have her in secret? Is it possible Q didn't know about her?
In several episodes of TNG, Spot (Data's cat) was stated to be male. Another episode shows Spot having kittens.
My first thought was that Data simply never checked to see if Spot was male or female, but that seems unlikely. Still, when I refer to continuity, I'm talking about things relevant to the franchise.....but I do find it odd that they'd miss that. (Again, a story line I don't recall. Sometimes I have to actually see an episode, or part of it, to remember it.)
First contact with the Borg? "Q Who" in Season 2 of TNG. BUT... Voyager comes along and the Hansen's (7 of 9's parents) 20 years earlier went on a mission to study the Borg.
Nobody knew her parents had been studying the Borg. They weren't part of Star Fleet, IIRC, so there was no "First Contact". Regardless, they were assimilated into the collective, so Star Fleet would never have known anyway.
In TNG, Scotty has no memory of Kirk's death, despite having been present on the ship where Kirk "died."
Can't explain this one...amnesia? LOL
There are many many more if you just google it.
https://www.trekbbs.com/threads/what-is-the-worst-continuity-error-in-trek-history.139195/ has a ton of them (and fan-made excuses).