Captain Kirk, Spock, Bones and Scotty all work the same shift. Who runs the ship during the graveyard shift?
The little known purple shirts
I don’t think it works that way. I mean, you rarely see the Enterprise attacked out of the blue - episodes usually start when they reach a planet, or encounter something in space, or receive a message from somewhere, giving the command staff time to assemble on the bridge. They probably take turns at the helm when they’re in warp or in orbit, but their main job is to call the captain when something happens.
Assuming you actually mean who runs the bridge, McCoy is irrelevant to that discussion.
Anyway, they don’t actually stand the same watches, at least not in routine flight; there are few episodes in which we see only Kirk or Spock on the bridge at the beginning and having the summon the other. (Okay, I admit it’s usually Spock on the bridge first; I’m specifically thinking of the Corbomite manuever.) Speaking only of the TV series, I should think that the conn (control of the bridge) rotates among Kirk, Spock, Scott, Sulu, Uhura, and a few others in four-hour shifts – again, in routine flight; helm, navigation, communications, science, the other stations would likewise rotate. But when they are nearing a destination that requires the “A” team, the officer at conn summons everyone else.
Someone will point out that we never see Uhura at conn. I reply that I am not responsible for NBC’s sexism. 
Anyway, she took the conn at least once in the animated series.
ETA: also, I’m not sure Kirk would ever actually stand a watch. He’s always on duty, and so can take time away from the big chair whenever he feels like it. He has paperwork to do, and he clearly sees his main duty as leading away missions anway, so he may take the big chair largely when they are half an hour or so from a destination that will require a captain’s presence.
Spock pulls double shifts, because Vulcans need less sleep than humans. That puts him on deck with Kirk when Kirk has the conn.
Third shift is taken by a no-name extra shavetail lieutenant, who has standing orders to not allow anything interesting to occur on his watch.
You know what you never saw on the Enterprise? Administrative work. I mean, running any organization involving several hundred people requires hours of meetings and acres of paperwork a week. Just once, I’d like to see Kirk sit in on a Payroll meeting.
This is the 23rd century. That’s what 22-year-old women with beehives, miniskirts, and go-go-boots are for.
Where would you cash a check?
Besides, I thought it was a no cash economy.
M’ress shaves her tail?
Isn’t it in “The Deadly Years” that Kirk signs some kind of report, and then later in the episode, after he starts to lose his memory, he asks a crewman to give him the report to sign and the crewman says he already signed it?
ST:TNG finally answered that question in their series. Og course, they had Data.
Don’t go there, Klimore. But waxing is more traditional.
Isn’t that pretty much every episode of Next Generation?
Not only Data, but there was one episode where Crusher had the conn and one where (shudder) Troi was on command duty.
There was also an episode where Riker was filling out forms for yearly crewman evaluations. Martok also tried to turn down a promotion because he didn’t want the extra paperwork that came with it.
TNG had those little clipboard-looking thingies for reports and signatures.
Kirk signed a lot of clipboards. I don’t recall Picard ever touching paperwork. Drank a lot of Earl Gray tea and polished his chrome dome.
In Voyager, it was shown a few times that Harry Kim was in command during third shift on occasion.
TOS had them also. In the early shows Kirk was often busy signing things before the big event hit. They went away when the writing got sloppy and they stopped treating the Enterprise like an actual navy ship.
On the other hand, they actually showed Picard’s office, which was clearly separate from his (enormous!) personal quarters. And Picard (particularly as the seasons progressed) tended to spend most of his on-duty time there. We just don’t see him poring over PADDs and his desk computer because, well, it’s boring.
Sisko also clearly did a lot of paperwork. I think he mentioned to some junior officer (Worf? Nog?) that they’d hate that part of the job when they got to be captain.