Star Trek, numerically sequenced

The Federation prides itself on the cultivation and propagation of a tolerant, all-inclusive and progressive society of equals. Because a great number of Starfleet Officers are either human (or humanoid-aliens), they are sometimes beholden to a nasty part of human-nature (or humanoid-alien nature) that insists on creating dichotomies between acceptable (us and people like us) and disgusting (them). Sometimes this makes for the thrust of a whole episode - the objectification of Data gets a bit of mileage - and sometimes a character makes an off-handed comment from the sidelines, briefly exposing a few prejudices yet linger. Kirk’s Klingon thing was effectively dealt with in ST:VI, to a point, so I’ll assume he figured it out.

5. Miles O’Brien on numerous occasions.
“The bloody Cardies can’t be trusted!”

While the good chief refrains from using the straight up derogatory slur ‘Spoonhead,’ there’s always something distinctly hateful in the way he says ‘Cardies.’ His struggle with the lingering racism of conflict is a part of his character, so there’s not much surprise when his old grudges re-surface, but considering the steps he took in ‘Cardassians’ and ‘Destiny’ toward not making snap judgments about the whole race, it’s always a bit disheartening to see the ‘common man’ slip back into old habits. He might not need the seminar, he knows what’s going on - “It’s not you I hate, Cardassian. I hate what I became because of you.”

4. Archer, as a matter of policy.

“Very disrespectful, but, boy, did it feel good!”
I’d rather not re-live Archer-diplomacy. 

3. McCoy to Spock, in Bread and Circuses (and most of the time)
“I’m trying to thank you! You pointed-eared hobgoblin!”
Oh, they love to joke around. They love to argue. But while McCoy’s inability to understand Spock (and vice-versa) is an integral and fascinating part of the original series (and a somewhat necessary audience-surrogation device), there’s really no need to resort to ugly slurs, Doctor.

2. Worf to Troi & Dr Crusher, in The Outcast

“That is a woman’s game…a man’s game has no wild cards.”
In an episode pulling the reverse allegory about gender roles, Worf is chosen to be the crew member who finds the J’Naii off-putting, and throws out this gem while playing poker with two women who were just talking about the equality of the Federation. And given other chest-puffing about Klingon women being ‘equals,’ Worf looks especially foolish. Worf was already booked for the seminar for the across the board racism culturally inherited from his Klingon-ness, but the sexism has him stay another day.

1.  Paris to Chakotay, in Caretaker
“Isn’t there some Indian trick where you can turn yourself into a bird and fly us out of here?”

Now, perhaps Tom thought Chakotay was a shape-shifting alien… but he uses the word ‘Indian’ and knows full well that Chakotay’s just a regular guy. Chakotay responds with a joke, as Native Americans have learned to do over 700 years (and counting, Voyager) of dealing with the great white jackass. Does nicely reinforce that Tom’s a real dickhole at this point.