All of the murders, really. This is commented on in Chapter 4 when Nagito discovers records of the first game's events.
In Chapter 1, there is a failed murder attempt using a knife stolen from a kitchen, and the Ultimate Lucky Student is the first suspect. The culprit also tries to talk their way out of getting punished as they both believe their murders are justified. Both execution cutscenes also begin with the agitated culprit being dragged out of the courtroom by the throat. Furthermore, the first victim is someone important from the story's narrative: Sayaka is the first (Decoydeuteragonist while the Ultimate Imposter is presented as Byakuya Togami from the first game. The culprits on the other hand were mostly comedic relief characters who got unlucky (Leon was targeted by Sayaka and had to react, while Teruteru overheard that Nagito was going to attempt to kill someone).
In Chapter 2, the victim is bludgeoned to death with a single blow. There are hints that a serial killer is responsible but that turns out to be untrue. Plus, the Panic Talk Action (equivalent to the Bullet Time Battle of the first game) isn't against the murderer, but someone trying to defend them. More subtly, the first murder of the first game has a baseball player use a sword to attack the victim, while this case has a swordswoman use a baseball bat to attack. Also, the dark pasts of both the victim and the culprit have been revealed in the case.
In Chapter 3, there is a double murder, and the first body found is actually the second to be killed. This victim is also someone who was deeply affected by losing their newfound friend in the previous chapter. The game also hints that two people are responsible (like the first game), but in this case it's a Red Herring. It also involves someone recovering their memories, and is the only killing that was cold-blooded murder, with no sympathetic or noble elements to it.
In Chapter 4, it is initially unclear on whether it is suicide or murder. The victim is the largest and strongest person in the class. The final words of the perpetrator motivate the students to keep living. Also, the culprit notes that there was a possibility of them being killed by their target instead, which would have led to a very confusing case, just like with Sayaka Maizono in the first game. Furthermore, both cases have been committed in an act of sacrifice.
In Chapter 5, the murder is staged with the goal of having a specific target (or in this case, targets) executed by the trial. The victim's death also involves the Spear of Gungnir, and the initially suspected cause of death is not what killed them. Also, both chapters feature a cutscene in which a bomb blows a character across the room (but doesn't kill them).
Chapter 5 also has a lot of parallels with Chapter 4 from Trigger Happy Havoc. Both deaths were initially engineered by the victims, and someone who isn't the blackened tries (and fails) to make the students lose the game and have everyone executed. The victim has several wounds that would have killed them before everything happened if not for their massive willpower (and luck), and it also comes off as a twist that the cause of death is poisoning. And as a little extra detail, both (ab)Normal Days/Daily Life sequences feature someone getting slapped for tormenting the future victim.
In Chapter 1, to provide a motive, Monokuma reveals that the sixteen students have had their memories at Hope's Peak removed, and comments that only a poor writer would wait until the finale to drop this reveal (which was what happened in the first game).
"Or, in other words... that one of us is a serial killer? That sounds like a plot twist from some kind of weird game or something..." referencing Toko's split personality as Genocide Jack.
The password the students were looking for ended up being 11037. This was deliberate on Makoto's part in developing the game as he wanted to put a failsafe password in to start the graduation program early in case Usami somehow screwed up the Hope Crystals plan, so he chose 11037 as it's very personal to him since that was what his childhood friend Sayaka left behind to exonerate him from murder. This comes up again when Hajime catches the fake Makoto in a snag when he doesn't know what the significance of 11037 is.
Nagito's suicide - stabbing himself, tying himself up, poisoning himself, and spearing himself, all inside a locked room setup - mirrors most of the DR1 deaths: Sayaka was stabbed, Chihiro was tied up, Sakura was poisoned inside of a locked room, and Mukuro was speared; additionally, numerous stab wounds on his thighs bring to mind Genocide Jack's habit of carving marks upon her leg after she kills someone, which is why Toko wears a long skirt. Seeing how he knows how the murder victims died, as he tells Hajime in Chapter 4, this is likely very intentional on his part.
At the end of the game, Hajime appears to converse with Chiaki's spirit. Afterwards, he gets white hair, red eyes, and a Battle Aura. This is basically what happened when Kiyotaka channeled Mondo's spirit in the first game, only this time it's Played for Drama.
Most of the characters are amnesiac members of Ultimate Despair, which is the same as the big twist of Danganronpa Zero.
A number of the presents directly reference previous material:
The gold-leaf practice sword that Leon used for self-defense against Sakura.
The crystal ball Leon used to activate the incinerator (stuck together with tape).
Ryoko's memory notebook.
Yuto Kamishiro's Hansel & Gretel bread bag.
A pressure point codex indicated to belong to the Ogami clan.
A knife said to belong to Mukuro Ikusaba.
A painting of Junko Enoshima.
A book describing a sword technique named after "Sayaka M."
In chapter 3, a sick Nagito will ramble about Ibuki having a twin sister, and how she's the "17th student", a nod to the plot twist of the first game.
When Hajime and Chiaki find Monokuma's poison inside Nagito's fridge, Hajime initially confuses it with protein powder, a nod to the poison/protein powder switch that was instrumental to the 4th class trial of the first game. The container even resembles the poison container from the first game.
The achievement for completing Byakuya's FTEs in the first game was "The Game Hungers". The achievement for completing "Byakuya"'s FTEs in this game is "The Game Hungers for Seconds... and Thirds".
