Which of these questions are correct correct?
Either of these only work if you are talking to a man (sir is for a man, and ma’am or madam is for a woman).
Assuming the person you are asking is a man or woman, these can be used. In correct written form, there should be a comma between the question (at the beginning) and the “sir” (at the end). The person that you are addressing is someone you respect, so you add the “sir” in to show that respect.
- Permission to carry on, sir?
- Permission granted, sir?
Note that I am assuming what you want to do is complete an action yourself. However, you are asking someone whether it is okay for you to do that first. The way that you have worded your question, though, leaves room for different interpretations.
For example, you can ask someone else to do something, or you can ask someone else whether or not you can do something. The above questions only work if you are doing the asking.
In addition, most people would not talk like this in real life. The only situation(s) that you might ask this is if you are in the military and your commanding officer has asked something of you. Otherwise, you would simply ask something like Should I go do… now?