As of asking this question, my knowledge of set theory is quite pedestrian. I've read about it in numerous nontechnical papers and even worked through three chapters of Jech - Set Theory, but in terms of what actual set theorists do I have no idea (my idea of forcing is people somehow shoving axioms into $\sf ZFC$).
I've heard many, many people talk about how $\sf ZFC$ is somehow wrong. Some examples:
I've read Wildberger's paper where he states the axiom of infinity as "There exists an infinite set" and proceeds to bash how informal it is. I've seen the MSE thread where this is discussed and seen arguments for why it is indeed not a strawman, but they didn't convince me. (He doesn't appear to have a problem with choice specifically, but rather with infinity to start with [and I guess without infinity choice isn't even a thing anyway.])
I've seen the last section of the paper on dividing by three (it's available online) where they talk about how choice is probably not true, without giving any mathematical support for this hypothesis. They even claim that there is work being done to show ZFC is inconsistent - is this still happening today?
I've also seen the Banach-Tarski paradox used as an argument against choice. Basically, they say that since choice on the continuum implies such an astounding result which is so obviously not true (why, exactly?) that it must be wrong. I find this argument similar to one in quantum physics - when Heisenberg proved the uncertainty principle no one panicked and seriously started talking about how quantum states must not form a Hilbert space after all.
This is about it. Are there any arguments with mathematical content against $\sf ZFC$ whether it be refuting infinity, choice, or something else?