I think this might have been brought up many moons ago, but I can't find a recent topic, so I'm bringing it up again.
Sometimes I think something is good to say, and then 10 minutes pass and I regret posting and want to remove my comment. I'll admit this is almost 100% an Anxiety issue, and therefore 100% a Me issue, but I can think of other reasons why someone might want to delete a post. Accidental double post - being able to delete your own comments in this case would save headache for mods. You asked a question, only to figure the answer out yourself shortly after. Or you're just an anxious mess like me and decide you don't want your post out there for whatever silly reason... D':
Other sites (Flight Rising, and I think Dappervolk also had this ability but the site's down now for beta so I can't confirm for sure) have this function and there doesn't seem to be much issue with it!

The only issue I can see is that people could use this to bump their threads by deleting their most recent post, and then making a new one. I'm not sure how people in general will feel about that, or if Staff can stop that from happening.
I'm uncomfortable with this. Other than in the news posts/comments, we can already go in and edit any post with no time limit. Which means, you can delete everything you wrote and replace it with "sorry, not what I meant to say" or "figured it out" or something equivalent.
I just feel that being able to completely delete a post would make it easier to cover-up rule breaking or get around the bumping rule as pointed out.
I'm also concerned about how this would affect achievements like post count and earning Forum Points - I go in write some spam, get my count up, possibly get an FP. Delete, repeat and thereby avoid getting warned for spamming.
In loving memory of Need posting achievements?Then join Posting Frenzy Achievement Items - searchable list List of Borders and cutouts New at Fresh and Flavorful Ping Group
No support at all. It would be way too easy for people to use this as a way to get away with breaking rules. You can already edit your posts if you’ve changed your mind about what you said, being able to delete them opens too much potential for rule breaking.
The rule breaking issue is actually, if I remember correctly, why this idea was rejected in the past. By allowing post deletion, it does make it far easier to break forum rules, as users could break a rule in their post, delete said post and say it never happened and therefore bypass obeying forum rules. I wasn’t a supporter of this idea in the past for that reason, still not now.
...these do not sound like arguments against the idea to me, but that is probably because when I think about "rule-breaking" I think about things people might do accidentally, like using Japanese words they think everyone on earth already knows in a post. Plus, editing would cover it up too.

I think if this were done in the right way, if possible, this would be a good idea.
Like if a deleted post is logged in the background, as I'd hope edited posts are, then if someone said something offensive/rude then deleted their post there would still be a log of it for rule enforcement. That is, of course, assuming edited posts are already being logged; if they aren't, then being able to edit posts are already bypassing the rules so this concern is moot.
Regarding bumping/spamming, would it be possible to delete a post but maintain a record that user x had previously posted? So you post, delete it, and try to post again and it says "you've already posted!".
I don't know if there's enough cause to actually code this, but if it wasn't overly complicated I wouldn't mind seeing this happen. I could see there being a few scenarios I'd use this.