Even though Spock and Uhura don't get as much screen-time, the moments they have are truly special.
A badly-injured Spock manages to track down where Uhura and the others are being held captive. The relief on her face when she sees him (after thrashing the drone mooks that were attacking him) is palpable.
Uhura:[worriedly as she tries to help him up] What are you doing here? Spock: Clearly, I am here to rescue you. [A smile starts to spread across Uhura's face]
What's even sweeter is that Spock told Jim he wanted to go down to Krall's area because Uhura was there. It's very subtle, but his facial expression and slightly-widened eyes when Chekov locates her necklace seals his determination to get down to the camp to see her. Also, McCoy promising to go with Spock to keep him safe.
The fact that Spock gave Uhura his mother's necklace.
At the end, it's clear that the couple are back together again—what with Spock telling Uhura that he enjoys her company and her calling Spock a "romantic" while touching the necklace.
Uhura: I thought you had to finish the mission report. Spock: I do, but I thought it would be more pleasing to engage with you socially. Uhura:[smiling] You old romantic.
And while the crew watch the Enterprise-A being built, she brings him a drink and he's looking at her with a small smile the whole time.
Scotty and Jaylah's blossoming friendship. He is able to convince her to go back to the "place of death" where Krall killed her family just by showing her a little understanding and offering the hope of strength in numbers.
Their relationship becomes even more heartwarming when Scotty tells her Captain Kirk pulled some strings to get her into Starfleet Academy.
They even appear to be flirting in the background as Kirk passes by Keenser at the party.
McCoy is a lot more understanding than one might expect of Spock's uncertainty, fear, and grief over Spock Prime's death.
When Scotty, accompanied by Jaylah, is reunited with Kirk and Chekov who were caught in Jaylah's trap, he greets the young navigator with a relieved hug. Chekov readily returns the affection with a joyful cry of "Mr. Scott!"
McCoy tries to counsel Kirk through his soul-searching in the beginning and shares some stolen booze with him.
When the Enterprise crew docks at Yorktown, Sulu is happily reunited with his husband and daughter.
Before that, the beginning shows that Sulu keeps a picture of his daughter at the helm.
In a way, after the reveal of who Krall really was, his greeting the sight of a now flying USS Franklin amidst the swarm becomes this. It's not unlike Kirk looking upon the Enterprise-A at the very end of the film.
Krall: ...My old friend.
The fact that Edison/Krall still vowed to keep Le/Manas and Woolf/Kalara alive and not absorb them.
After we discover Krall is in fact Balthazar M. Edison and Manas is Anderson Le, one of the two other crewmembers of the Franklin who survived, the fact that Krall did not immediately leave Manas to destroy Yorktown and then that Manas handed his old captain the weapon and told him to go is both heartwarming and slightly disturbing.
At the end, the surviving Enterprise crew celebrates Kirk's birthday.
In the beginning, when Kirk decides to apply for a promotion and a desk job, who does he decide should become captain in his stead? Spock.
Then in the end, after having spent a great deal of time considering just what kind of man he is and thinking about leaving command, he reaffirms his position as a captain instead of taking on a promotion and being assigned to a desk. A move even Commodore Paris clearly agrees with.
The iconic Opening Narration being delivered at the end by all seven main characters as they prepare for the voyages of the Enterprise-A.
Kirk: Space, the final frontier. Spock: These are the voyages of the Starship— Scotty: —Enterprise. Its continuing mission: McCoy: To explore strange new worlds. Sulu: To seek out new life— Chekov: —and new ciwilizations. Uhura: To boldly go where no one has gone before.
Jaylah's face when she finally makes it off Altamid.
Spock looks through Ambassador Spock's belongings and finds a picture of the Prime Timeline crew (circa Star Trek V, from what would be about 25 years into the future at that point). Consider that Spock Prime wasn't planning on permanently leaving when he got stranded in the past, so for him to have this picture, it means he must've always been carrying it with him at all times, no matter where he went. Consider he was, at that time, likely the last surviving member of the original Enterprise's crew.
Which makes Kirk Prime's line from that particular film, "I've always known I'll die alone," a lot Harsher in Hindsight.
Bones (both as a doctor and a friend) fussing over Spock's injury, clearly being very concerned.
When Bone's jokingly says he's going to throw a party when Spock leaves Starfleet, Spock starts laughing. McCoy does not, because he realizes that Spock is in shock and he needs to get him back to the ship to get medical attention fast.
Kirk and Spock's brief moment in the lift, as both plan on telling the other of their decision to leave the Enterprise, with Kirk applying for the admiralty and Spock's plan to take over Spock Prime's place on New Vulcan after the latter's passing, only for both to decide to keep it to themselves until after the mission. The fact there is no snark on either side, with genuine feelings of friendship, shows just how close the two are compared to the last two films. Carries on at the end as both now somewhat realize the other's decision, only for both to deny anything of the sort, having now decided to stay together.
Kirk giving a brief speech to the entire Enterprise crew on what could be his final mission with them, thanking them for being such a wonderful crew and for staying with him in all those years of deep space exploration.
A bittersweet example as Edison had always believed he and crew were abandoned by the Federation, a belief that fueled his anger and desire to destroy the Federation. At the end, it was revealed that the Federation has always seen Edison as a hero who helped make it possible due to his actions during the Xindi and Romulan Wars. Doubles as a tearjerker as Edison would latch onto his hatred to the bitter end.
After Scotty preaching about the importance of loyalty among a crew, Jaylah gets to see it soon afterward. Even though Manas tries to demotivate her by asserting that the Enterprise crew will leave her, she looks down and sees that Kirk is still circling around on the motorcycle trying to get to her, even though everybody else is gone. This seems to motivate her to keep fighting, knowing that she is not being abandoned.
The entire film does a great job of honoring the late Leonard Nimoy throughout the film, but one particular moment near the film's end shows the young Spock going through the deceased ambassador's belongings, and he comes upon an old photo of the TOS bridge crew. What's particularly heartwarming is that it's from Star Trek V: The Final Frontier, which has infamously been viewed as the weakest film in the franchise, especially for TOS.
The registry number of the USS Franklin is NX-326, a direct reference to Nimoy's birthday.
The end credits has two title cards labeled "In loving memory of Leonard Nimoy" and "For Anton."
The many references to the era of Star Trek: Enterprise are heartwarming for fans of the oft-ridiculed series.
The Beastie Boys are noted fans of Star Trek and essentially stopped as a band after the death of MCA/Adam Yauch in 2012. Aside from serving as a Call-Back to the first Kelvin Timeline film, the use of "Sabotage" is a fitting tribute to the fans: His music saved the day. Also, while they don't license their songs out as a general rule, they made an exception for the franchise since they were fans and were given context for how "Sabotage" would be used.
McCoy: Is that classical music? Spock: Yes, Doctor, it would seem to be.
These two official cartoon posters from the official Japanese Star Trek Beyond blog: Pining Spock and Theater.