Axiom of Choice has many variants like the followings:
There is a choice set for every family of non-empty sets.
All sets are well-orderable.
Of course in many cases one don't need AC to prove existence of a choice set or well-ordering for a particular family or set. In such cases one can prove these facts in ZF.
Question: What are examples of set theoretic axioms which violate the axiom of choice badly? I mean examples of axioms which imply non-existence of choice sets (or well-orderings) on almost all families of non-empty sets (or sets) which existence of a choice set (or well-ordering) on them is not provable in ZF.
As it stated in comments, $AD$ is certainly a classic example of an anti-choice axiom. However I would like to know about other variants of such axioms and also the intuition behind them which illustrates why these axioms are violating $AC$ badly not merely implying $\neg AC$ which is possible just by producing a single set which has no well-ordering.