I don't think we really know anything about the backstories of most of the TOS crew other than Kirk (genocide survivor; watched half his crew get killed by a Dikironium cloud creature), Spock (estranged from his family), and a tiny bit about McCoy (divorced, not that that's really "traumatic"). SNW later establishes that Uhura's parents died in a shuttle accident (I think they got that from the novels), but the rest of the crew is a blank slate in terms of backstory.
On TNG: Picard (shitty home life, abusive father, later elaborated on Picard's series into a literal gothic melodrama complete with suicidal mentally ill mom (ugh.)); Riker (mother died young, father was a dick, on his one from age 15); Data (planet destroyed by his brother; abandoned by creators; forgot his early life; went 26 years before he got his first friend. Not really traumatic though, on account no emotions); Worf (family annihilated by Romulans); Troi (domineering mother; dead father; dead older sister she never learned about until adulthood); Crusher (widow; parents killed in childhood; also that whole "generational alien sex ghost" thing we're not supposed to talk about). Geordi, though, explicitly had a good childhood (notwithstanding talk in the writers' room of retconning him to be a literal Rosemary's Baby)
On DS9: Sisko (watched his wife die at Wolf 359; that whole prophetic birth thing (ugh.)); Kira (child soldier during an attempted genocide); Bashir (basically rewritten by his own parents); Quark (everyone loves Moogie, but she is a pretty horrible mother); Odo (medical experiments); O'Brien (war-related PTSD); Dax (repressed serial killer memories); Garak (there's literally an entire novel about how shitty his childhood was)
On Voyager: Janeway (no trauma that I recall); Chakotay (ex-terrorist; I guess his home planet was taken over by Cardassians? They never really say); Torres (abandoned by her father; resentful of her mother; resentful of her own biology and cultural heritage); Paris (shitty father); Neelix (genocide survivor); Seven of Nine (holy shit, just everything that's ever happened to her). I can't recall Tuvok having had anything traumatic in his past, but I very much interpret "Meld" as showing that he's kind of a psychopath at heart and only a razor-thin veneer of logic prevents him from being a serial killer; Harry Kim had a happy childhood, notwithstanding an attempt to give him the "Rosemary's Baby" backstory that the writers had originally intended for Geordi (almost like even in the 90s, the writers couldn't think of how to develop characters without giving them trauma); Kes...had a childhood that lasted, like, a month or two, so it might not count
On Enterprise: Um...yeah, okay, I don't know this one as well. Archer's dad died before his ship could be built, which is...bad, I guess. T'Pol...um, I guess she went to that jazz club one time and it freaked her out. Reed's parents seemed awful in that very "upper-class British" way. Um... Yeah.
On DIS: Burnham's family was killed by Klingons, and then her school on Vulcan by blown up by terrorists, so, you know...trauma two-fer, I guess; Saru was from a species that was being systematically culled before puberty; Tilly's mom was neglectful; um. Book had an abusive dad; Adira saw their lover killed in front of their eyes; Owosekun was disowned by her luddite commune for joining Starfleet. Um. Airiam lost her husband and all her skin in a shuttle accident; Bryce (or maybe Rhys? I don't remember) was in a hurricane as a child; Rayner's planet was occupied by the Breen; probably some of the other bridge bunnies exposited trauma at some point during the fourth season, I don't remember
On Picard: Hoo-boy. Raffi was right about the attack on Mars, but lost her family and developed a drug problem because of her obsessive conspiracy theorizing; Rios had his father figure enact an illegal order and then kill himself in front of him; Elnor had his father figure abandon him (what the fuck, Jean-Luc); Soji's entire life was a lie, except her twin sister (who was killed); Jurati...actually seemed to have a pretty good childhood, but made up for it by having an incredibly traumatic present; Shaw has oodles of survivors guilt over Wolf 359; Jack Crusher, Jr....doesn't actually seem to have had anything particularly bad ever happen to him, but still acts entitled to run around with a giant chip on his shoulder (god I hate him); and then there's added trauma for established characters, like: Picard left Starfleet and bears the weight of the Romulan supernova; Troi and Riker lost their son because of the synth ban (and then, according to season 3, Troi telepathically suppressed Riker's ability to mourn (ugh.)); Seven of Nine was betrayed by her lover and had to mercy kill her adopted son; Crusher...got pregnant and randomly decided that her traumatic backstory was so overwhelming that she simply had no other choice but to keep her son from his father and ghost all of her friends (ugh! ugh! ugh!); something is implied to have happened to Worf, but they never said; Geordi...still doesn't really seem to have a lot of trauma, though the official tie-in materials say that he was in charge of Utopia Planitia when it got bombed. [Okay, so maybe there's a bit too much trauma on this series.]
Lower Decks: Mariner lost her best friend; Rutherford had his entire life deleted by an evil admiral; Shaxs has so much PTSD from the resistance that he'll go spare on anyone who even mentions Bajor; Billups has a shitty mother who keeps trying to trick him into have sex; um...that's kind of it.
Prodigy: They're all child slaves, first of all. On top of this, Dal was sold by his other mother figure (and doesn't even have parents); Rok (had to play the monster in an arena as a small child); Zero (separated from their hivemind; Jankom was an orphan sent off on a sleeper ship; and then woke up before everyone else and had to ditch the sleeper ship; Gwyn was raised to be a weapon by a cruel, domineering father. Murf seems to radiate peace, though.
Strange New Worlds: Pike knows his future is kind of crappy; Una lived her entire life hiding what she was; La'an was bullied for her family connections, and then had her family ate by Gorns; M'Benga, Chapel, and Ortegas have war trauma; and then there's the previously mentioned stuff with Spock and Uhura.
So yeah; going through it, I think that the only clear outliers in terms of trauma are Picard (way too much of it) and Enterprise (not enough).
