How to ask about one's availability? "free/available/not busy"?

Solution 1:

All three variations of the sentence you give are grammatically correct, and are virtually (but not completely) equivalent. There are a few points worth noting in particular.

1. It is far more conventional to say "let me know when you are busy" than "let me know when you are not busy". The negative is considered redundant, and they are understood to mean the same thing, i.e. they are both enquiring about availability.

2. Saying free or available rather than busy may be considered a more "positive" enquiry. It may also simply mean that you expect the person to be busy rather than free, rather than the other way round.

3. Saying available rather than free is considered slightly more formal, though I wouldn't worry much about usage cases. (Most people wouldn't think twice if you used either in a formal or colloquial situation.)

Solution 2:

All three of those are equally appropriate, and as far as I know, completely equivalent. The only thing I would change would be to say "in more detail" rather than "in more details".