I have seen every episode at least half a dozen times over the decades and I am pretty sure nobody smoked; not even some exotic alien hookah or something.
In the days when cigarette ads were all over the place (in the 1960s) on TV and in print and characters did smoke in movies and on TV, if indeed this was never shown on STOS, was it a conscious decision? Or did it simply look too low tech for the show?
Not the Original Series, but in Star Trek V: The Final Frontier, a Federation ambassador named St. John Talbot is seen smoking on screen. This was a film with the Original Series cast in the Original Series timeline, so we see that smoking in that time frame was canon, though the smoker was not in Starfleet. In fact, the character was portrayed as weak and corrupt, so the cigarette is probably part of that commentary. Here's a photo:
According to this source, smoking occurred in a number of the movies and other series, but never in the Original Series. The closest TOS got to showing smoking was a scene with a cigar humidor and an ashtray on an alien's desk.
Gene Roddenberry was adamant that there not be any smoking on the show, despite pressure from the Network and sponsors, which included cigarette companies.
"I was uncooperative in other ways too. For example, no-one aboard the USS Enterprise smoked, and tobacco advertising was key to network profits in those days"
Yes, but it has been excised from all videos.
I recall seeing a brief scene (probably in the episode teaser - the opening scene before the main title) where Kirk and Spock are walking in a corridor (presumably to the transporter room), when Kirk says something to the effect of, "Crewman! Put out that cigarette!".
This was back in the 1970s and I've never seen it since. I thought it was in either "Charlie X" or "Tomorrow is Yesterday".
But, the "The Official Star Trek Trivia Book" by Rafe Needleman (1980), in question #250 asks, "In which episode do you find someone smoking on the Enterprise? To which he gives the answer, "A Piece of the Action".
I know the book has errors (for example, it states Ophiucus VI was Mudd's original destination in "Mudd's Women" - when it was Ophiucus III), which is why I am not convinced because it disagrees with my memory.
So, yes, there was a single (now excised and memory holed) scene in Star Trek The Original Series where someone was smoking a cigarette onboard the USS Enterprise.
UPDATED: 21 August-2023
In the episode "Patterns of Force", a woman is briefly seen who is holding a short (empty) cigarette holder (around 32:20):
In the episode "Spectre of the Gun", some men are seen smoking cigars (around 13:05):
