Functional Magic: All the serious contenders for power have it, generally either in the form of the Pattern or the Logrus. Some have additional sources.
Healing Factor: A very slow version for Amberites: they can recover from almost anything, but major damage (like paralysis, or blinding) can take months or years. A much quicker variation for shape-shifters.
Berserk Button: Seen in the second book. For some reason, Benedict strongly disapproves of discussions of the various family claims to the throne. It's implied that he's simply sick to death of the family habit of plotting, hence his self-exile to get away from everyone.
The deaths of his shadow retainers is enough to drive him into a rage to kill his younger brother.
Big Brother Instinct: Not very strongly. But he won't allow his younger siblings to kill each other in his presence.
Cain and Abel: Zig-Zagged. It would take a powerful incentive for Benedict to decide to kill one of his siblings...
The Chessmaster: Played with. He's generally considered an unbeatable strategist, to the extent that Corwin says if Benedict wanted the throne, Corwin would bow down and pay him homage because there would be no way to stop him. But Benedict doesn't care to turn those abilities against his siblings.
Cool Horse: A black-and-red-striped horse named Glemdenning, which seems virtually fearless - earthquakes, forest fires and so forth don't seem to faze it.
Fatal Attraction: Sleeps with the hellmaid Lintra, who happens to be commanding an army invading his realm, and who later cuts off his arm.
Four-Star Badass: Partly through innate talent and partly through obsessively acquired experience.
Handicapped Badass: Events prior to Benedict's first appearance in The Guns of Avalon cost him his right forearm. Although he shares the Healing Factor of his family, it works slowly enough that his appearances in all future books invoke this trope.
Hot Skitty-on-Wailord Action: Played with. Benedict (great-grand)fathered Dara with a creature of Chaos, the latter of which is a shapeshifter. Dara is said to be "the first of [her] line to bear all the markings of humanity," but whatever Lintra and her fellow hellmaidens were, she was probably close enough to not be this trope with Benedict.
Immortal Immaturity: Played with. He still can't cope very well with family, but at least he treats ordinary Shadow beings like people, not toys.
The Juggernaut: Do not piss off Benedict. Ever. He has spent countless lifetimes fighting wars to perfect his skill as a general and is nigh-invincible with a blade. He will not listen to your stammered excuses, and he will cut through trees to get at you quicker.
Master Swordsman: Even more so than the other Amberites to whom this applies. Although Benedict is The Juggernaut in a fair fight, some of the talents ascribed to him by fans and in the RPG push him into Memetic Badass territory - his ability to parry the bullets of invisible snipers, for example.
Memetic Badass: Benedict is of course an epic-scale in-universe example of this, as seen above. Or with all the characters saying that if he wanted Amber, they'd all just have to roll over and submit immediately.
Corwin: "I fear Benedict. [...] He is the Master of Arms for Amber. Can you conceive of a millennium? A thousand years? Several of them? Can you understand a man who, for almost every day of a lifetime like that, has spent some time dwelling with weapons, tactics, strategy? [...] All that there is of military science thunders in his head. He has often journeyed from shadow to shadow, witnessing variation after variation on the same battle, with but slightly altered circumstances, in order to test his theories of warfare. He has commanded armies so vast that you could watch them march by day after day and see no end to the columns. Although he is inconvenienced by the loss of his arm, I would not wish to fight with him either with weapons or barehanded. It is fortunate that he has no designs upon the throne, or he would be occupying it right now. If he were, I believe that I would give up at this moment and pay him homage. I fear Benedict."
It gets to the point that, in the third book, when it becomes a possibility that one of his siblings is the Big Bad, Corwin has a short inner monologue to the effect of, "I really, really hope it's not Benedict, because if it is, I'm screwed." Later on, Fiona says the Big Bad can't be Benedict because if he'd wanted the throne, he could have simply taken it through military might and strategy, and nobody would've been able to stop him.
Noodle Incident/Remember When You Blew Up a Sun?: The Moonriders of Ghenesh were the only serious threat to Amber in her entire history (prior to the Black Road) and their invasion force made it as far as the slopes of Kolvir itself. It's implied that they overpowered Oberon himself, Eric, Corwin, Bleys and all the other Princes of Amber. Benedict was recalled and stopped them cold with an elite regiment in the pass above Arden. Corwin fanboys this.
Only Sane Man: He willingly exiled himself for centuries because he knew his siblings would not react responsibly to Oberon's absence.
Papa Wolf: Subverted. In the second book, Corwin thinks he's seeing this - he's actually seeing a more conventional Roaring Rampage of Revenge. Played straight in the fifth book.
Sole Survivor: Not just during his centuries of military adventure off in Shadow; he's also the only survivor of Oberon's first set of children. This may explain some of his other traits, considering that the family scheming he despises was what got his two brothers killed.
Random(describing his battle with a mighty Shadow serpent): "Benedict would not have missed the eye. He would have had one in each pocket by then and be playing football with the head while composing a footnote to Clausewitz."
An exception to the general tropes listed above: Bill is just an ordinary human attorney on 'our' Earth whom Corwin knows fairly well. Corwin tries to keep Bill uninvolved with his abruptly far-more-complicated life, but to no avail.
Four-Star Badass: Subverted in the first book when he and Corwin are summarily overwhelmed by Eric's generalship: played straight later on, when Corwin is told Bleys deliberately lost that battle in order to get Corwin out of the way.
Heel-Face Turn: One of the three who initially ally with Chaos to get Oberon out of the way.
Karma Houdini: He's included in the general amnesty in the fifth book.
Name of Cain: Well, duh. He ends up as the only prince to actually kill a brother with his own hands.
The Unseen: during most of books three through five.
Coral (Princess Coral of Begma)
Coral is a princess of Begma, a Shadow-realm close enough to Amber to trade and otherwise hold commerce with the city. Thus, not all of the tropes apply to her. Well, actually, only Cain and Abel and Immortal Immaturity are disqualified, and even those may just be due to her not being immortal for long enough yet.
Badass Bystander: Coral starts out in this role and rapidly becomes far more central to events.
Took a Level in Badass: Dworkin tends to Coral's eye injury by implanting the Jewel of Judgement in place of her missing eye. Coral immediately starts displaying unusual powers similar to those Brand and Corwin had when they attuned to the Jewel. Unfortunately, Author Existence Failure intervened before the reasons and consequences of this surgery could be explored.
Corwin (Prince Corwin of Amber; Carl Corey)
A God I Am Not: One of the few family members to take this attitude... in his humbler moments.
Anti-Hero: Generally a Type III. And this is nicer than he used to be. Stories about his past imply he used to be IV/V.
The Atoner: Having used the terrible power of a blood curse to hurt Amber in his darkest moment, Corwin spends the rest of the first series trying to make it right.
Attending Your Own Funeral: Well, Visiting Your Own Memorial Tomb. Corwin quite likes it up there. And then he takes delight in pissing on his own grave, mainly because how many people even get the opportunity?
"...Then he had me blinded and sent to the dungeons."
He leaned forward and studied my face. "Yes," he said, "I had heard that. How was it done?"
"Hot irons," I said, wincing involuntarily and repressing an impulse to clutch at my eyes. "I passed out partway through the ordeal."
Fate Worse than Death: What's intended for Corwin when he's blinded and tossed into the dungeons to be forgotten except for once a year when he's paraded around as a trophy of Eric's rule. Later Corwin is told that this torture was "for his protection," although his source had good reason to cover up the details.
...it was not long before I overtook him, riding as though he were pursued by the Devil, which he was. I spoke not a word when I unhorsed him, nor afterward, and I did not use my blade, though he drew his own. I hurled his broken body into a high oak tree, and when I looked back it was dark with birds.
Unreliable Narrator: Corwin narrates the first five books and demonstrably gets some things wrong, either through ignorance or misremembering. Also, his descriptions of his siblings are slanted through his personal prejudices.
Victory Is Boring: Corwin found this out after Eric's death, and soon decided that becoming King wasn't for him after all.
Warrior Poet: Literally. A renowned swordsman and poet in Amber. On Earth, Corwin was both a soldier and a composer of popular music. His narration exemplifies the trait as well, switching from hard-nosed, cynical commentary to prettily-expressed, poetic sentiment.
He is said to have composed "The Ballad of the Water-Crossers," the anthem of Amber's merchant navy (the most important aspect of their culture).
Tsundere: She shows her attraction to Corwin by challenging him immediately to a fencing match. If one assumes she really cares for Corwin at all, then her true personality reflects this trope even more strongly, to Yandere levels.
Dierdre (Princess Dierdre of Amber)
Badass Princess: At one point in Nine Princes in Amber, she and her brothers are attacked by werewolves. She is unarmed, so she grapples one and snaps its spine.
Damsel in Distress: When we first meet her, Deirdre is tied to a tree and has to be rescued (although she joins the fight once released). At the end of the series, she is held hostage by Brand. This time it doesn’t end well.
Faux Action Girl: During the battle at the Courts of Chaos, she is the most heavily armored of the princesses and wields a large axe. Two minutes later Brand is holding a knife to her throat.
Debatable, as Brand can take on Benedict. Dierdre may not have had an opportunity to defend herself - if, for instance, she accepted a Trump contact and he overwhelmed her as he once did to Martin.
Hot Skitty-on-Wailord Action: Dworkin claims that he fathered Oberon with the Unicorn. Since he's a shapeshifter, it's possible; but he's also notably mad and nearly as big a compulsive liar as the rest of his family, so who knows.
Bastard Bastard: Eric, bastard son of Oberon and Faiella. Mortally wounds his brother Corwin and carries his comatose body off to some plague-infested Shadow. Later, he locks Corwin up and burns his eyes out with a red hot iron (Though to be fair, the blinding was Julian's idea.)
Hot Witch: Her vanity is alluded to in Corwin's narration.
Karma Houdini: She was part of the cabal that intended to use the Courts of Chaos to unseat Oberon. On the other hand, she exerted herself pretty heavily to defend Amber, so her pardon might be justified.
Dumb Muscle: While he is not dumb per se, his trusting and straightforward nature makes him appear like this in comparison to his scheming relatives. On the other hand, he's not usually considered a threat, so nobody goes after him.
Good Is Not Nice: In Sign of the Unicorn, Gerard suspects Corwin of fratricide and treason. Gerard takes him to an out-of-the-way place, knocks him silly, dangles him over a thousand-foot drop, and delivers a Badass Boast.
Gerard: "You may be as innocent as you say or as guilty as possible[...] Look down at the black road. Death is the limit of the distance you travel if that is your doing. I have shown you my strength once again, lest you have forgotten. I can kill you, Corwin. Do not even be certain that your blade will protect you, if I can get my hands on you but once. And I will, to keep my promise. My promise is only that if you are guilty I will kill you the moment I learn of it."
Super Strength: True of all the Princes, but Gerard is acknowledged as the very strongest. Corwin, who is capable of using a recliner as a lethal throwing weapon, regards Gerard's strength as "a thing of legend."
To the point where he pulls a chain that had been successfully restraining one of his brothers for years out of the wall it was anchored to and uses it as a weapon.
At one point Merlin, having taken an interest in computer design (after his phases of interest in sorcery and Trump artistry), picks a particular Shadow where the physical laws will allow him to design a machine programmed like a computer, but capable of designing 'virtual' Trumps instantly and thus reaching through Shadow. Soon, it develops its own personality and agenda. Subverting the usual tropes, Ghostwheel doesn't turn into the Big Bad of the series. Since it's an "ideally synthesized" being, usually only present as an immaterial projection, and quite young, it doesn't share in the tropes listed at the top of the page aside from Functional Magic.
Fiona: "Julian has no friends. That frosty personality of his is warmed only by thoughts of himself."
Cool Horse: The grey stallion Morgenstern - seemingly tireless and fearless, capable of astonishing speed and agility. It may be a horse-shaped construct rather than simply an extraordinary animal: Corwin recalls at one point that Julian created it.
Morgenstern stood five hands taller than any other horse I had ever seen, and its hooves were like polished steel and its eyes were the dead color of a Weimeraner dog's.
Deadpan Snarker: He doesn't like Corwin, or much of anybody else, but he's too fond of his self-control to be aggressive about it.
Julian: "We spend so much time lying to each other that I decided it might be amusing to say what I really felt. Just to see whether anyone noticed."
It's Personal: When Caine's body is found, Julian insists on the privilege of questioning and executing his killer.
Jerk with a Heart of Gold: Eventually, although it’s possible he was this all along and it's only Corwin's perspective that's changed.
Corwin learns that Julian suggested to Eric that Corwin have his eyes burnt out and be imprisoned as... a sideways way of protecting his life.
Saving Corwin's life was more for Eric's benefit than Corwin's, according to Julian himself - otherwise, had Oberon had ever returned to reclaim the throne, having Corwin executed would have been Eric's only unpardonable act. Julian still qualifies for the trope in the above example, but from a slightly different angle.
Bitch in Sheep's Clothing: We find out at the end of the series that he actively tried to magically enslave Merlin, and when he found out it hadn't worked he attacked Merlin and tried to forcibly put the spells back. He accepts his defeat rather gracefully, though.
Evilly Affable: Although sinister and Machiavellian, Mandor is usually polite, and even capable of friendship. Just don't count on him allowing your friendship to get in his way.
A cryptic, masked sorceror who - for reasons unknown to Merlin - keeps trying to kill him. And keeps sending him flowers. As a Shadow being, Mask isn't subject to most of the 'universal tropes' above - except for The Chessmaster and Functional Magic.
The Casanova: Merlin seems to be setting himself up for a Love Dodecahedron with Coral, Julia, Gilva, Rhanda, etc. as possible paramours. He generally doesn't say no to a willing woman, much like his father and grandfather.
Disappeared Dad: Merlin was raised knowing his father was technically his enemy. Later becomes Disappeared Dad in a more literal sense, because nobody seems to know what happened to Corwin after the fifth book.
My Beloved Smother: Dara has definite plans for Merlin, having had him for just such a reason, and doesn't mind using a few dozenhundred political assassinations and mind control of her own son to have her plans come true. Also to a lesser extent, Merlin's aunts are very snoopy in his life, although they generally seem more benevolent.
Person of Mass Destruction: Merlin is normally quite low-key. But if you push him hard enough, you'll discover that under that calm, reasonable exterior is a talented sorcerer who's good at improvisation, can call on the two greatest sources of power in the known universe, can kick the ass of most beings even without magic, and as a last ditch option, can summon elemental chaos to utterly obliterate everyone and everything within the target area.
The Casanova: Oberon seems to leave kids everywhere he goes, though considering how long he's lived, the actual number seems relatively small. He also seems to have trouble with the "forsaking all others" part of marriage. In the Merlin books, his ghost claims to have fathered forty-seven children (thanks in part to time proceeding more slowly in some shadows than others). That doesn't sound like a lot when you've been alive for maybe tens of thousands of years, but of his children (who have probably at least as many years between them) only four have had children of their own that we know of, and in three of those cases magic might have been used to help things along.
He was a little guy, maybe five-six in height, weighing perhaps one thirty-five. But he sounded as if he were talking dead serious talk. I felt reasonably sure that he meant it when he said he'd take on two or three bruisers, singlehanded.
You Killed My Father: He eventually decides to be satisfied with the amount of revenge he's achieved to that point, so the trope is partially subverted.
Ambiguous Gender: "It" and "its" are used. Possibly later subverted... if Dworkin's claim that the Unicorn is Oberon's mother can be trusted.
Big Good: Played straight in the first series; toyed with in the second.
Crystal Dragon Jesus: Averted. The later books reveal that there is a religion based around the Unicorn, but the royal family seem to regard it as a respected totem creature rather than actually worshiping it.
An exception to the usual tropes (aside from Functional Magic) because Vialle is not a member of Amber's royalty except through marriage. She is a minor noblewoman of Rebma, a sculptress and a seer despite being blind.