Insults and abuse don't belong in this wiki. Confronting bad ideas and faulty reasoning are encouraged, but personal attacks have no value. They are just noise.
So, whenever you see insulting language, it is acceptable to just delete it. There is no need to explain yourself nor to seek approval from other wikizens - just do it. If you think it necessary, you can note that you made a RudenessDeletion, but that tag often just leads the offender to add more insults.
Don't DisagreeByDeleting. The goal here is to remove noise that does not clarify the issues, not to remove alternative ideas. If the author makes interesting points along with the insults, try to leave the points intact.
If the insults are good-natured, humorous, or valuable in some way, there is no need to delete them.
"Insults and Valuable"? Did someone DisagreeByDistorting?
There are many shades of gray, and lots of room for misinterpretation. For example, if somebody writes "What you are saying is nonsense!", some will consider it an insult, whereas others will consider it a request for clarification. A statement like "C++ is a horrible language. Why do all these lemmings use it?" will be seen as an insult targeted at people who like C++, and not simply an opinion about the C++ language. It is best to AssumeGoodFaith and to be tolerant of others' opinions and the ways they express those opinions. And if your own well-intentioned writings are deleted because they offended somebody, consider ways to restate your point.
There may be cases where it is valuable for the community to discuss an individual's shortcomings (for example, grossly inappropriate behavior, a pattern of abusive behavior, dishonesty, etc.). In such cases, the issues should be presented on a page separate from other topics, and accusers should be careful about any unwarranted attacks. Personal insults on such a page are candidates for deletion just like anywhere else.
Please bear in mind that as well as deleting insults, you can also make them (although somebody else will probably delete it).
But, what if they put their name beside their words?
After giving this quite a bit of thought, I am inclined to agree that it is desirable to delete insults. I have a lot of reasons for leaning in this direction and since I'm sure its not an exhaustive list, I thought I would start listing mine here, and request others to add to it. (It may even make sense to cluster the items under the headings of Pros & Cons).
Pros (for deleting insults):
Alternatives to DeleteInsults
See also RudenessDeletion, DontDeleteFlamebait, WelcomeToWikiPleaseBePolite, CriticalSpirit