Alas, Poor Scrappy: While Detroit-era Vibe wasn't exactly popular, his death is so graphic and brutal most people tend to sympathise with him.
Angst? What Angst?: The current version of Johnny Quick apparently had a pretty horrific past that he is very keen not to talk about. His psychological scarring comes out in his willingness to kill people and laugh about it.
Asspull: The Justice League of America once went up against Despero (an alien with vast mental powers, at the time recently powered-up to be almost as strong and invulnerable as Superman) with a mostly C-List team. How to beat him? With an innate Martian power that the Martian Manhunter had never before mentioned that he had, because using it was so stressful that any given Martian can only do it once in their life. Despite that, it was still a fairly well-received story. Another weird power of his was the ability to see through the flow of time. This somehow resulted in him being immune to the powers of an opponent who completely rewrote reality. Pulling never before seen powers out of his ass is the Martian Manhunter's shtick. And then getting lit on fire the next time said power would have been useful.
Author's Saving Throw: For a few years, Prometheus was portrayed as much less competent than in his first appearance and spent most of his time as a hired goon for other villains. This was Retconned into being Prometheus' wayward apprentice, who had stolen the real Prometheus' gear while he was locked up physically and mentally.
Considering that in said first appearance (written by his own creator) Prometheus' biggest triumph was murdering Retro (who wasn't an actual super-hero, just a JLA fan who entered a contest) before failing to complete his takeover of the League Moon Base and getting whipped in the groin (literally) by Catwoman....well, his own competence is more touted than verified.
Felix Faust is one of the most power hungry sorcerers alive. Kicking off his career by selling his infant son's soul for his own ability, Faust committed many acts of cruel murder and torture to fuel his black magic. When he finally realized he could no longer bargain with his own tainted soul, Faust resorted to harvesting the soul of an innocent little girl to trade to demons. Faust offered his assistance to Black Adam to help resurrect Black Adam's wife Isis, but deceived Adam into thinking Isis's revival had failed, while Faust kept a paralyzed Isis to serve as his personal sex toy.
Grant Morrison's JLA: Earth-2 brought us Brainiac's Antimatter Universe counterpart. An organic syntellect who was captured by Ultraman and forced to be his slave, Brainiac schemed to not only escape his bonds, but to upgrade himself into an "Nth Level Intelligence." In order to do this, he swapped a plane in the Antimatter Universe for one in the Matter Universe, killing all the passengers on both in the process, and triggering a war between the Justice League and the Crime Syndicate. While the teams were busy fighting, Brainiac set in motion a plan to merge the Matter and Antimatter Universes, a move that would have resulted in the annihilation of all reality, plotting to use the energy from the resultant explosion to complete his upgrade and become a god. Willing to destroy all life in order to better himself, Antimatter Brainiac was every bit as vile as his Matter Universe twin.
Dork Age: The Detroit-era is generally (though not universally) held as a low point in the League's history.
Tomorrow Woman of Grant Morrison's run on JLA. Originally debuting and making a Heroic Sacrifice in the same issue, she was so popular she reappeared in a special one-shot dedicated to her, was briefly brought back in Hourman #2 by the main character, and then was brought back in Trinity, where she was permanently revived as a human woman with superpowers after the series ended.
Once he was retconned out of being a founding member in the New 52, Martian Manhunter defaulted into this role. It's almost impossible to find a discussion regarding the New 52 League and not have someone bring up wanting J'onn back to replace Cyborg.
Foe Yay: Max's charmingly creepy JL:GL "I miss you" monologue to Booster Gold, delivered while lovingly beating Booster half to death with a pipe.
Fridge Horror: JLA: Rock of Ages showed a possible future where Darkseid ruled. Darkseid had a scar on the right side of his face in that future and his jawline was more pronounced. In JLA: World War III General Eiling says that he is "the face of tomorrow". Note that he had red eyes, a lantern jaw and a scar on the right side of his face. Now take into account Final Crisis, where Darkseid manages to cheat death by hiding in human hosts...
Heartwarming Moments: V1, #158. By this point, Red Tornado (who happens to be an android,) has been a member of the group for a while, but still has doubts about both himself, and his relationships with his other teammates. There are times when he isn't sure if his teammates truly trust him; where he wonders if all they think of him is just some machine. In this issue, that matter is solved once and for all. After Red Tornado contains a freaking explosion, he briefly falls unconscious, and drops into the sea below. He probably could have flown out shortly after, but Batman, after dealing with other matters, flies back to Reddy in his jet, and pulls him out of the drink with a rope. As he's climbing, we get this:
Red Tornado: "But... how could you find me, out here...?
Batman: "How do you think? I followed you in my Batplane! A Justice Leaguer doesn't desert his friends, Reddy!"
Red Tornado:"Am I... your friend?''"
Batman: *Batman, of all people, simply clasps Reddy's hand, and smiles; Reddy smiles in turn.*
Max Lord's like the embodiment of this. In the first issue of Generation Lost alone, he comes back from the dead to mind-control two cops into shooting each other, beats Booster Gold half to death with a pipe, infuses himself with the blood of a small army of random extras, wipes the memory of his existence from the minds of *everyone in the entire world* (except our small band of plucky heroes), and rounds it all off with a glass of claret and a cigar on the battlements of Checkmate HQ. Magnificent.
Prometheus's got no superpowers, but he's stampeded through the Justice League three times now, and has pulled off ridiculous stuff like completely disabling the Flash by lying to him, or talking Superman into suicide, or blackmailing the League into letting him go after destroying Star City. He even teamed up with Luthor once (they caused World War III, no big deal), which led to a funny moment when Prometheus showed Luthor around his "crooked house" and the two masterminds passive-aggressively sniped at each other.
Memetic Molester: Aside from all the Ho Yay going on between him and Booster, in Generation Lost #9, Max Lord gives Magog a very creepy wink while talking about his 'cojones'.
For Max Lord, it's killing Ted Kord. He crossed it again by forcing Magog to kill himself. After playing with his head enough to get him to attack the heroes. It's implied he was influencing Magog from Generation Lost #4 up.
Narm: The circumstances causing John Stewart to join the League as Hal's replacement. See, Hal was unable to attend a summons. Was he held up with work? Personal problems? A super-villain attack? His duties as a Green Lantern? Uh... no. He slipped on a bar of soap in the shower, because he wasn't looking where he was standing.
Vibe, a member during its "Detroit" era, was widely hated for being a "hip", slang-spouting caricature of Puerto Rican youth whose power was basically super-breakdancing. His inevitable death, though, is one of the saddest moments in the League's history. Ironically enough, in the New 52 continuity, he's been given a big push by the company and is already enjoying more popularity than his original ever did.
Fans and editorial alike loathed Triumph. According to creator Christopher Priest, they missed the point that he was supposed to be a jerk.
Vibe and Steel from the reviled Justice League Detroit were killed off several years after they were created due to negative fan reaction. Decades after their deaths, the characters are almost never mentioned in-universe unless in a negative manner. This itself got lampshaded when Black Lantern versions confronted the surviving Detroit Leaguers and basically asked why they were remembered as jokes while their teammates, Gypsy and Vixen, got to join the "real" League.
Triumph, in almost any appearance not written by Priest.