Classical Mythology: It has a large influence in this universe, especially in Gaea, where it is ruled the Greek Pantheon alicorns.
Expy: Zeus is one of the Zeus from Disney's Hercules. In addition, the as-of-yet unnamed necromancer Big Bad from Velvet Sparkle and the Queen in Stone is pretty much the Myths and Birthrights version of King Sombra. Word of God outright says so.
She started trying to atone again after Twilight's Awakening.
The Theocracy: Equestria is literally this due to being directly ruled by alicorns.
A Dream of Pride and Envy tells us that topics which contradict the common believes in Equestria involving Celestia (such as she having a sister and cousin, and that she was close to death after her battle with Amon during the Second Battle of Airagos Forest) are considered blasphemy.
World Tree: Ioka has a sacred tree found in the Taiga.
Myths and Birthrights tropes
Adorkable: The first chapter shows that Twilight also has a checklist for symptoms of her usual panic attacks.
Even as a goddess, Twilight still acts like her usual eager-to-learn student self.
Badass Boast: Given one by Shyara's mother ( Astraea) to Ares in chapter 6 during Shyara's dream:
Astraea: No, Ares, you will not have them. With the last of my strength, I spit upon you. With the last of my will, I curse you. With the last of my love, I will save my herd. With the last of my magic, I will send them somewhere you can never reach them.
Badass Grandpa: The old zebra shaman Zubu is very elderly, yet can still kick Gilda's hind whenever she displeases him.
Comfort Food: In chapter 3, we are told that Luna ate a tub of mint and chocolate ice-cream to calm her nerves once when the stars started shunning her after returning from her exile in the moon.
Cool Sword: Celestia's sword, Coronal Edge. For added coolness bonus points, it was apparently forged on the surface of the sun and appears to have a will of its own.
Crapsack World: Gaea is a parallel world which has been destroyed by a catastrophic war and slaughter between Alicorns and Titans.
Culture Clash: The alicorn fillies (especially Tyr) saw a big difference in cultures between their world (Gaea) and the world they have been transported to (Ioka).
There were few instances of cultures clashing during Twilight and her friends' visit to the Zebrican Empire, especially ones which involve The Empire's laws and morals.
Deity Of Pony Origin: Fluttershy was never born as an alicorn but in chapter 16, Artemis gave her her alicorn powers which ascended Fluttershy into an Alicorn of Forests.
Also in the same chapter, Aphrodite exorcised Serene by forcing her powers to be given to Rarity instead, turning her into an alicorn.
Divine Date: Fluttershy and Big Mac's romance from interlude 2 and afterwards.
Friendly Enemies: Captain Hardy and Bloodrose 'Bonnie' Belle ( who is Rarity and Sweetie Belle's mother) are enemies (Hardy is a navy officer and Bloodrose is a pirate captain), yet very polite towards each other.
Ghost Memory: What Fleur, Rarity and Fluttershy were having in earlier chapters due to having their souls merged with the souls of their Alternate Universe selves.
Good Is Not Nice: Faust is a good example of this, not only did she attack Zeus and Hades unprovoked, but she says that she's glad that the griffons are endangered because they used to be savage killers. She's also not afraid to say things like "You disgust me!"
Aphrodite isn't exactly the nicest pony around, either.
Good Parents: Derpy, Twilight's foster parents, and Rainbow Dash boasts that she will be this when she realizes that she's pregnant.
Knight Templar Parent: Astraea. The alicorns foals are in danger (she thinks). She must protect them, and she'll do anything to protect them.
She's like this to stars too. She believes that Twilight stole them from her (even if she's in Ioka and not Gaea) and she is trying to regain her physical alicorn body to usurp Twilight and retake the stars, which she believes are hers and only hers.
Literal-Minded: In Interlude 2, we are shown that Sol (the Sun) can be dense when following orders. Cue the Sun literally dropping from the sky when Celestia ordered it to the first time.
Mama Bear: Twilight to the stars. When three stars went missing, she took her time to find them again. She had almost slain Draco the Star Dragon for devouring them (plus mortally injuring Pinkie Pie).
Mood-Swinger: Iridia has been described as one in Book 1: Chapter 9. River remembering her queen in her hyperactive mood on a mountain is one such example.
No-Holds-Barred Beatdown: Trixie gives one to the Slaughter Daimon that attacks her, Shyara, and Rarity, literally beating its face in with a rock. To say that it doesn't work would be an understatement.
Obfuscating Stupidity: Blueblood normally acts like a very pompous and snobbish prince, though we are shown in chapter 17 that he is the current Vice Chief of Naval Staff for the Crystal Navy, in other words, a Spymaster.
Perpetual Frowner: Zubu. His frowns and scowls are more frequent than his smiles. Though, that could be because he believes that he's the Only Sane Man (zebra, actually).
Sirius. She's very hard to please and impress. She also doesn't accept Twilight as her new mistress unless she gives her a reason to respect her.
Gilda, especially after her cousin's death.
Privateer: The Sea Serpent's crew are privateers working for Prance to pillage Trotugese merchant ships.
Serious Business: While Luna can't see Fluttershy's date with Big Macintosh as very important compared to Fluttershy's magic lessons, Iridia thinks that her date is very important.
Sharing a Body: Rarity, Fluttershy and Fleur each have a dead alicorn from a parallel world living inside their body. Fluttershy's was willing to fade to Elysium to let Fluttershy live, but Rarity's and Fleur's were battling for dominance. Rarity seems to have lost the battle for total body control.
The Dreaded: Chapter 6 tells us that Iridia had titles which makes her seem to deserve this trope (Deathgiver, Black Star, She Who Shall Not Be Named and Bloodmaned).
Zeus, of course. When Shyara found out he was on Ioka and in Canterlot, she was first afraid of him being here, then she was afraid of him not being on Gaea due to being this trope to keep the others in line.
The Empire: We are getting hints that Zebrica could be considered one, even having a tyrannical Empress.
The Obi-Wan: Zubu is an old zebra shaman who is also being a mentor to Gilda after finding her wounded after a large battle against rival griffin aeries. He patiently tries teaching her some defensive zebra spells, but he is starting to become less patient considering he is nearing death of old age, especially when he sees her practising offensive spells instead of shields.
Your Normal Is Our Taboo: Tyr told Fleur de Lys casually that Celestia ordered that anyone who impersonates a priest is sentenced to an execution. Other ponies, especially Celestia and Rarity, found that information horrifying. The reason Tyr said it was because she thought Fleur is impersonating one of their priests (she didn't know she was in a completely different world).
The Longest Night tropes
Been There, Shaped History: During The Longest Night, Twilight and Luna both went back 1000 years because of Celestia's wish at that time after Luna/Nightmare Moon's banishment ( the 'shooting star' was actually Iridia as told in chapter 6 of Myths and Birthrights) and talked with Celestia, which in turn gave her hope and made her prepare for Luna's return.
Velvet Sparkle and the Queen in Stone tropes
Absurdly Youthful Mother: In chapter 3 of Velvet Sparkle and the Queen in Stone, Velvet had birthed her first daughter, when she was 16.
Though she had excused herself that it was a different time and she was living and grew up in a different culture.
Badass: You'd be lying to yourself if you don't admit that Velvet is a badass when she lived with the Halla in Velvet Sparkle and the Queen in Stone.
She is also the mother of a half-Halla who is a legendary White Hind, a prince of the Crystal Empire and she is also the foster-mother of the Goddess of the Stars.
Black Magic: The Dark Runes that Velvet carelessly used out of desperation in chapter 5 of Velvet Sparkle and the Queen in Stone.
Cool Sword: Llallawynn, Velvet's, and River Sparkle's, sword. It was found in the bowels of Gamla Uppsala by Velvet in chapter 5 of Velvet Sparkle and the Queen in Stone and has a star's spirit inside it. It is also enchanted that only Velvet and anyone in her line can wield it, others would only find it as dull as if it is made of stone.
Going Native: Velvet Sparkle in the side-story Velvet Sparkle and the Queen in Stone. She was found orphaned in the Taiga by the Halla and lived with them. After a few years, she considered herself as one of the Halla.
Obviously Evil: Tyr and Shining Armour think that way about Sombra when Velvet talked about him in chapter 4 of Velvet Sparkle and the Queen in Stone.
Our Zombies Are Different: Draugen in the Taiga are undead mummified Halla from the times when the Halla had burial ceremonies.
Raised by Natives: In Velvet Sparkle and the Queen in Stone, it is shown that Velvet Sparkle lived and grew up with the Halla (deer-folk in the Taiga) when she was a filly after losing her family in the Taiga. She also mated with one and had a daughter with the Halla.
Really 700 Years Old: 2,000 actually. The old pegasus that Velvet met in chapter 5 of Velvet Sparkle and the Queen in Stone was actually Algol, a fallen star.
Ungrateful Bastard: Sombra in chapter 4 of Velvet Sparkle and the Queen in Stone was very angry at Velvet, Growler and Sylph because of their help in defeating the draugen he was fighting.
Demon Lords and Archdevils: Amon in A Dream of Pride and Envy was a shadow-like demon lord who battled and almost killed Celestia during the Second Battle of Airagos Forest.
Forgone Ending: Most readers of A Dream of Pride and Envy are also readers of Myths and Birthrights, so most will know what will happen to Namyra.
Mama Bear: Iridia. Namyra's death caused her to start a very long winter that killed countless ponies.
The Pacification of Crystal Valley tropes
Badass Princess: Cadence managed to unite two enemy legions from the War of the Sun and Moon to wipe out the Diamond Dog infestation in the Crystal Valley.
Enemy Mine: The 3rd Lunarius Legion and the 2nd Harmonius Legion, who were enemies in the last war, had cast their differences aside and teamed up under leadership of Cadence to defeat the Diamond Dog Packs in the Crystal Valley.
Turned Against Their Masters: The sword Penumbra used to be owned by Nightmare Moon who had personally forged it to kill Celestia. It eventually rebelled against its mistress and bonded to Cadence instead.