You need to identify the objective severity of the insult. Being called names, compared to animals, sworn upon, etc. - these would be severe insults, that should be reported to the management.The responsibility of every adequate management is to not allow such insults. You should also address such insults immediately by politely pointing out that the comment is unworthy, unprofessional, and insulting. You don't want to insult anyone back, because you want to be professional. Also, public insults should be addressed publically. A public apology should do for an early stage. If management or team tolerates direct insults against you, you are in a toxic team and should consider switching jobs. Don't pursue this path ("war-path") unless yo have really been objectively insulted.
Then, there are statements, that can be insulting, that don't contain the direct swearing or names, but that just paint a bad professional picture of you. Like, "He will break it if you allow him to touch the database." and stuff like that. For these remarks, you need to be extra careful. Here are some tips:
- Trust your colleagues to know you and see your work to judge whether doubt about you can be bad.
- If the comment is mistaken, counter-argue and politely state your proficiency in the matter. "Sad to hear you doubt me. I am quite experienced with databases."
- If the comment IS true, you may want to be humble. "I am still learning databases, but I am careful enough not to do crazy things."
In any case, the general rule is that you want to react immediately, react correctly, indicate that you noted the tone, for example. "I appreciate your comment, Steve, but I do find the wording somewhat offensive".
Your philosophy is that you should be thick-skinned as not to allow the comment to set you off and ruin you, but you also must show the borders of due conduct by addressing those cases and showing that you see and guard your borders, gently, but firmly. It is also a signal to other chat members - they feel compelled to note and possibly, react. This also is a possible log to show to leadership is this escalates.
Then, there is a third type. When people are just careless, or distressed, and you are pretty sure they are not really going after you. In this case, yo may want to ignore a comment, especially if you have reasons to believe that the person is not really attacking you consistently.