This is basically a long comment. I like <a href="https://mathoverflow.net/a/437202/290">Nik Weaver's</a> 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 <a href="https://mathoverflow.net/a/437200/290">brought up</a>. 
 * 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 *<a href="https://arxiv.org/abs/0905.1677">The concept of a set</a>*, which I personally found quite eye-opening.