Othello - Most number of legal moves in a given turn
It is very hard to say it without browsing all possible combinations. Another question is are you asking about moves in game that make sense or any random game.
I was playing this game many years and usually one consider move choosing out of 3 maximally 5 possibilities for good move. However in mid-game there are 10-20 legal moves frequently. 

You can try some maths to estimate max number of legal moves. Board is 64 fields. You can move only on adjacent. So in my opinion max number possible moves is when you have filled center 6x6, lets say white are on boarder of 6x6 and black are in center. Black have a move so he can move on any field left on board. It gives 28 possibilities. I would say it is maximal estimate.

If you still need to analyse that you should have a look on that program. It is used by most good players. There is also big openings book.