10. Kirk Is Not Dead - Amok Time
To begin this list with an oldie - this moment succeeded where, unfortunately, the JJ Abrams films did not. Leonard Nimoy's Spock epitomised the cold, logical depiction of Vulcans early on in Star Trek. All of the race that followed were based in part off of his portrayal.
From his initial appearance alongside Jeffrey Hunter in The Cage, where the character had not yet been truly defined, Nimoy began to establish who this race were. Zachary Quinto, while a worthy successor, was stymied with a script that had him break this coolness too soon. The entire point of Spock was that he didn't lose his control.
In the second season episode Amok Time (yes, that's the music that's running through your head right now), Spock is overcome with the need to....go home.....and as such his control over his emotions slips more and more as the episode goes on. The ship races back to Vulcan and the now-famous fight between Kirk and Spock takes place, horns blaring, shirts tearing. Spock naturally wins, but with a twist.
The pure moment of joy that erupts when Spock is overcome with his delight in seeing Kirk alive and well at the end is one of the most smile inducing moments that Star Trek has to offer. That's the key element - it was wonderful because of its spontaneity, not in spite of it. It would be a long time again before we saw Spock lose control again, making this piece of joy special in the history of Star Trek.