I'll have responses to individual posts in this thread in a bit, but one thing it's important to note is the difference between character development and character arc. Briefly, character development helps to explain who a character is at a particular point in time in the story. A character arc, in contrast, is the narrative arc which examines how a character changes over the story. You need character development to have a character arc, because if you don't clearly know who a character is, you can't determine who they are changing into.
I would say that Discovery had very little of either one of these outside of Micheal Burnham - and in that particular case both the characterization and the arc was very muddled and confused.
As was noted in another thread, any dialogue in a show is put there for a reason. Generally speaking the reason is one of two things - either to move along the plot, or to give us further insight into the character. Thus one way to determine the "character moments" in Discovery is to subtract all of the plot-critical discussion and see what's left. There are a few notable examples of pure character-related dalogue - Stamets brushing teeth with Culber, Tilly and Burham talking about burritos, etc - but these moments were for the most part in the first half of the season, and fell to the wayside for pretty much pure action once Discovery went to the Mirror Universe.
As for character arcs, It's very hard to say. I guess Saru was a coward until he wasn't. And Stamets was a grump until he wasn't. And Tilly was socially awkward except when the plot required otherwise (like in Madness...). None of this came across so much as character growth as it did retooling the characters, or perhaps even just inconsistent characterization - because very little evidence was given onscreen for why the characters may have changed.
Regarding Burnham, she got the most development, but her arc became pretty tortured as the series went on. Act 1 seemed to point to a straight-ahead arc for her. Basically the prologue implied that she acted the way she did ("mutiny" and all) because of fear of the Klingons, brought on both by childhood trauma and being raised in a shitty way by Sarek. As a result, she was under the false belief she was acting logically when she was really making gut-level emotional decisions. Act 1 slowly shows her getting out of her shell. By Into the Forest I Go it seemed like she had conquered her fear of Klingons, facing down Kol (almost too easily it seems). Then Act 2 is basically emotionally torturing her. Worse yet, she makes the rash decision to save MU Georgiou for no other reason than because she looks like her dead mentor - again making a split-second decision on emotion. The season ends with her again "mutinying" and making more rash decisions (trusting Georgiou not to kill her, and L'Rell to go along with her plan) only now hey, it works, and everyone gets a medal. The only realization/growth seems to be her understanding that she was racist towards Klingons due to childhood trauma - a thread that DIS totally dropped after the prologue only to come roaring back in the back half of the last act.