Yeah, this…I just went through the TOS Season 1 episode list…
The Enterprise’s initial duty in each episode
The Cage: Exploration
Where No Man Has Gone Before: Exploration
The Corbomite Maneuver: Exploration
Mudd’s Women: Police/Military (Maybe…I’ve not seen the episode lately, so I’m not 100% on why they were chasing Mudd at the start.)
The Enemy Within: Scientific
The Man Trap: Humanitarian/Transport
Charlie X: Scientific/Transport
Balance of Terror: Military
What Are Little Girls Made Of?: Scientific/Transport
Dagger of the Mind: Transport
Miri: Humanitarian
The Concience of the King: Complicated, but the lie told to trick Kirk into maybe performing police duties is humanitarian.
The Galileo Seven: Humanitarian (the Enterprise’s main duty) and Scientific (why the Galileo splits off and gets into trouble)
Court Martial: Military, nominally, though it’s not clear what the Enterprise was doing when the accident happened.
The Menagerie 1&2: Another court martial, so again military, nominally, but again, not performing military duty.
Shore Leave: Shore leave. So, none of the above.
The Squire of Gothos: Unclear duty, looks to be transport.
Arena: Military (though, ironically, the actual fighting of the episode takes place outside the context of formal military action).
The Alternative Factor: Scientific
Tomorrow is Yesterday: Resupply, none of the above.
The Return of the Archons: Scientific/Historical/Something of that Sort (determining the fate of a ship that crashlanded 100 years before)
A Taste of Armageddon: Diplomatic (and accidentally caught up in a bizarre war they had no idea about)
Space Seed: Unknown duty.
This Side of Paradise: Scientific
The Devil in the Dark: Humanitarian (while they intend to kill the Horta, they think it’s a pest-exterminating job, not a military/police action, as they don’t know the Horta’s sapient - and, of course, when they determine this, they switch to diplomacy)
Errand of Mercy: Ironically…military.
The City on the Edge of Forever: Unknown duty.
Operation – Annihilate!: Humanitarian/Scientific
So, a count (due episodes like Conscience of the King, this count is slightly fuzzy):
Military/Police: 4-6 episodes
Exploration: 3
Other Scientific: 7
Humanitarian: 5-6
Diplomatic: 1
Other/Unknown: 4-6
I don’t feel like going through the other seasons or series, specifically, but, discounting stuff like the latter seasons of DS9, where they were actually in the midst of a full-scale war, and thus military efforts came to the forefront, the count will be similar - if anything skewing a little harder toward ‘unknown/other duty’.