Great question. ASAP is one of those dangerous phrases that should be used with caution. It never be used with anyone "higher" in status. I.e. never say to your supervisor or boss: Please do this ASAP. And if you are a good boss or supervisor, don't use it with your employees! It usually comes across as mildly rude. You can use it as an inclusive again with caution. "Team members, we have this important request. Can we get this done ASAP? Thanks." In speech it can be even ruder! "Fred, I need this ASAP" is fine if you and Fred are equals and friends. Otherwise, it's pushy. Of course, you will supervisors and bosses using this all the time, but I prefer to work with my employees not order them about. There I hope I answered your question ASAP!

It is fairly informal. You'd use it in a message or email to a colleague, but not in formal correspondence.

ASAP is informal; used in personal email or internal memos. You spell it out in full in formal situations otherwise you are seen as a lazy person - not good in the eyes of your boss. It's like Japanese, you speak differently to persons of different status.