This is basically a long comment. I like Nik Weaver's careful distinction between the question of whether ZFC is consistent and the question of whether it is (e.g. arithmetically) sound, and would like to further propose a refinement of the question to not just whether people think ZFC might be inconsistent or unsound, but rather: if you were told that ZFC was inconsistent or unsound, which axiom would fall under your suspicion as something to be thrown out first?

• Finding replacement suspect is something that's already been brought up.
• A finitist might find the axiom of infinity suspect, but probably not anyone else.
• Of course there has always been an air of suspicion around the axiom of choice. Personally I do not believe that e.g. non-measurable sets exist in any reasonable sense so I am sympathetic to this sort of thing.
• Nik Weaver has argued against the power set axiom, e.g. in The concept of a set, which I personally found quite eye-opening.

This is basically a long comment. I like Nik Weaver's careful distinction between the question of whether ZFC is consistent and the question of whether it is (e.g. arithmetically) sound, and would like to further propose a refinement of the question to not just whether people think ZFC might be inconsistent or unsound, but rather: if you were told that ZFC was inconsistent or unsound, which axiom would fall under your suspicion as something to be thrown out (edit: or weakened) first?

• Finding replacement suspect is something that's already been brought up.
• A finitist might find the axiom of infinity suspect, but probably not anyone else.
• Of course there has always been an air of suspicion around the axiom of choice. Personally I do not believe that e.g. non-measurable sets exist in any reasonable sense so I am sympathetic to this sort of thing.
• Nik Weaver has argued against the power set axiom, e.g. in The concept of a set, which I personally found quite eye-opening.