In my (limited) experience of algebra, I have only seen a handful of axioms used. The wikipedia page on outline of algebraic structures lists a table of the following axioms for structures with one binary operation. The only axiom that I can think of that's missing is idempotency.
I personally find it surprising if there are so few natural axioms for binary operators (6 basically). Are there other axioms on binary operators that are used? Perhaps in much more niche contexts? I would be interested to see a larger list of axioms and to know in what contexts they would be used.