If you see something that violates the Code of Conduct you should flag the post for moderator attention as already suggested by David Carlise in the comments. You can either choose rude or abusive as reason or give a more detailed account in the generic in need of moderator intervention. It may or may not help to state matter-of-factly what particular behaviour or language you found offensive or problematic in the comments to make the poster aware of the issue (especially since there is no auto-notification for rude flags, cf. Is underlining the reputation of a just-do-it-for-me asker inappropriate?, though I assume people would be notified if their answer is edited). But you should not feel compelled to do that and comments run a risk of producing long discussions and arguments that go nowhere and result in personal grudges.
The code of conduct is pretty general and many terms are open to a wide variety of interpretation. Subtle put-downs or unfriendly language are concepts with a lot of wiggle room and subjectivity.
Remember that people here are from very different backgrounds and cultures. Some people may be from backgrounds where directness is valued, while others prefer a more indirect and subtle approach and perceive directness as outright rude. Furthermore, not everyone's mother tongue is English, so communication may not always be as smooth as one would hope. People may not pick up nuances that their words might have for other people and might not realise connotations and more hidden meanings that some phrases might carry (see also Ulrike Fischer's comment). Non-verbal components like tone and body language are absent in written communication, which can make nuances even harder to pin down.
Given all that, I suggest to give people the benefit of the doubt and suggest not to assume bad intentions where other explanations are equally plausible.
As for people 'showing off', again there is a lot of subjectivity involved in judging those situations. If I ask a question, I hope that someone may come along to show their skill and answer my question. So surely some kind of showing off is welcome, expected and desired. When people share code that goes beyond what was asked in the question, improves on related or unrelated matters or approaches the question from an entirely different angle that need not mean that they just came here to brag, it can also mean that they have the sincere intention to help me make my code 'better' (easier to understand, maintain or extend; free of possible negative side effects or interactions; ...). Apparently 'improved' code in answers is a source of frustration for some (see also What NOT to do when you're trying to help a newbie?), but it is important to understand that wanting to show off one's impressive skills is not the only possible explanation for improving code.
It may help to take a step back and remember that improvements and criticism of code I wrote is not intended to be an attack against me personally nor should it be seen as such. Of course I have invested time to write my code and I am associated with it in some way, but if I see every suggestion to make my code better as a personal insult, then it can become hard to get help and improve. That is not to say that people should be able to say all kinds of unfriendly things and then make excuses along the lines of: 'But I didn't say ... about you, I said that about your code'.
A perceived showing off effect may also occur when the answerer misjudges the expected level of expertise. An answer that over-explains things may look patronising and can be seen as showing off ('look at all these concepts that I know so well that I now explain all of them to you'). An answer that under-explains and uses many concepts that the person who asked the question may not be familiar with can equally look like the answerer is trying to show off. Balancing these extremes is an art and those who answer questions need to balance not only the interest of those asking the question (whose level of expertise might be hard to suss out), but should ideally also take into account the wider set of possible future visitors and readers.
So again I believe that in general there are good explanations why people's posts can come across as showing their expertise even without them posting solely to brag about their impressive skills. Therefore I suggest again to give people the benefit of the doubt and suggest you don't assume people just want to show off and brag and have no intention to help you. (And even if you sincerely believe that people are just showing off, I don't think it makes a lot of sense to call them out on it. After all, you'll have a hard time coming by good evidence that it is true and 'accusing' people of just showing off can be seen as an unnecessary personal attack. At least I would find it hurtful if it was suggested that I just want to brag when I'm sincerely trying to help.) If people over-explain, don't see it as an insult and try to let it slide. If people don't explain certain things you don't understand or use concepts you have never heard of, feel free to ask in a comment.
If you just think that an answer is bad, you should feel free to cast a downvote. Again it may or may not help to explain your decision in a short comment, but you should not feel compelled to do so if you don't want to, even if people who are aware that a downvote happened comment about it and ask for an explanation.