Does anyone still seriously doubt the consistency of $ZFC$? As someone self-taught in set theory beginning with Donald Monk’s excellent book on MK set theory, $ZFC$ has always seemed like a weak set theory.
Despite this, the majority of professional contemporary mathematicians still seem to view it as a very powerful theory, providing more than is necessary for their work.

Are there any professional mathematicians who seriously doubt the consistency of $ZFC$, and if so, why?

I wrote a few papers this year proposing set theories much stronger than $ZFC$, and as the year draws to a close I’m finding myself taking the consistency of $ZFC$ for granted as a ‘religious belief’. Are there any heretics present?

EDIT: The responses to this question have all been fascinating, and amount to one of the best christmas presents I’ve ever woken up to. Thank all of you, and merry christmas!
 A: Some more detail about Nelson's work:
For most of his life, Ed Nelson tried to show that Peano Arithmetic (and weaker systems) was somehow inconsistent.
This was (initially) based on the idea that one could take the enumeration of all partial recursive functions and somehow make this construction total, thereby deriving a contradiction (via a diagonal argument).
Ed later refined this to an attempt at showing that the exponential function led to inconsistency.  This was motivated by the Bellantoni-Cook framework of safe recursion in which the exponential function stood out for being the 'first' function with exceptional properties.
A: When I was a graduate student Jack Silver was famous for trying to refute first, the existence of measurable cardinals, and second, the consistency of ZFC. His attempt at refuting measurable cardinals led directly to the theory of ordinal indiscernibles over $L$ and $0^\sharp$, which are now a core part of our understanding of the nature of the constructible universe in the presence of large cardinals. What an amazing and valuable contribution to the subject he had made that way, even though he had never achieved his initial goal of refuting measurable cardinals.
And although I have heard many people speak against Silver's inconsistency research program, my view has always been:
Main observation. We cannot show that Silver's ideas about inconsistency are incoherent.
Because of the incompleteness theorem, we know that even if ZFC is consistent, then it is consistent with ZFC to hold that they are inconsistent. Thus, Silver's world view will be as consistent as his opponent, and there will never be an argument against the view that ZFC is inconsistent that does not beg the question by assuming that this theory is consistent or something stronger.
Meanwhile, of course, many set theorists, including almost every large cardinal set theorist (but not quite all, like Silver), believe not only that ZFC is consistent, but that consistency rides much higher in the large cardinal hierarchy. There is no proof, for the reasons I gave above.
Another part of my view, however, is that it is precisely because we cannot prove that ZFC and large cardinals are consistent that we are interested in them. Namely, Gödel had identified with his theorem the incompleteness phenomenon, and from that we know of the existence of a tower of theories standing transfinitely above any foundational theory we might entertain. We know there is a tower of stronger theories above whatever theory $T$ we might have. One way of constructing such a tower of theories is to form the theories $T+\text{Con}(T)$ and $T+\text{Con}(T+\text{Con}(T))$ and so on.
But how remarkable that the large cardinal theories themselves instantiate the predicted tower of increasing consistency strength. These theories are not formed by some trivial closure under consistency statements, but rather express profound statements of infinite uncountable combinatorics. These axioms thus have independent interest, and yet fulfill the predicted tower of consistency strength.
So we are glad to have found in the large cardinal hierarchy the predicted tower of consistency strength, and it is not worrisome that we cannot prove consistency — rather, this was just the feature that we were seeking.
Disclaimer. Although I heard Silver talk about set theory at length, having taken several of his graduate courses, I never once heard him discuss his inconsistency research program. All my knowledge about it is second-hand, from other senior set theorists whom I have no reason to doubt. I suppose that in light of the extent of opposing views, it might have made some sense or was at least understandable for Silver to keep his research program relatively private. But actually I wish he had given a series of talks about his research program and his views on the matter — that would have been fascinating! (From my current perspective, I chalk this up to the differing cultural norms about the tolerable levels of disagreement in mathematics vs philosophy.)
A: For decades I was not particularly suspicious about the consistency of ZFC but I was rather surprised about how it had become the standard choice when it contained axioms such as Replacement, which seemed to be unused by all mathematicians other than logicians. Relatively recently I learnt how to use interactive theorem provers, and the prover I use uses a dependent type theory as its foundations. My impression of dependent type theory is that it doesn't have any "exotic" axioms, and so it always felt to me to be "much more likely to be consistent" in some sense: it just seemed to have the bare minimum to do mathematics and nothing more. So I was rather surprised to learn later on that Lean's dependent type theory is equiconsistent with "ZFC + for all naturals n, there exists n inaccessible cardinals", a theory manifestly stronger than ZFC; whilst I would imagine most mathematicians are happy to believe that ZFC is consistent, some do express concern about large cardinals. So for me this felt like another version of the "Zorn's lemma is clearly true, but the well-ordering principle is clearly false" (or whatever the saying is) dichotomy: when presented in the right way ZFC can look even more "obviously consistent". In some sense this other viewpoint on ZFC made me a really solid believer in its consistency, because type theory actually models mathematics in a way far more closely aligned to how I mentally model it (e.g. I don't actually believe that $\sqrt{2}$ is a set, and no doubt Gauss/Euler didn't either).
One final comment regarding large cardinals -- people seem to be nervy about them because existence of an inaccessible implies ZFC is consistent, which can't be proved within ZFC by Goedel. However as Jeremy Avigad pointed out to me, if we're doing mathematics in ZFC (as many of us are -- I don't use Lean's higher universes for example) then we are implicitly assuming it's consistent (because otherwise we'd be wasting our time) so why are we fussing about people assuming something we believe anyway?
A: We have enough experience with ZFC that I think we can say it is consistent with some confidence.
This isn't too surprising --- I don't know if there's a way to say this precisely, but morally speaking consistency is "easy". Any collection of axioms is inconsistent only if the negation of one of them is a provable consequence of the others. Thus for a system to be inconsistent there must be some special interaction between the axioms, and again, I don't know how to say this rigorously, but to me that says consistency is "normal" and inconsistency is "abnormal".
What people who ask this question sometimes miss, though, is that there is a difference between consistency and soundness. ZFC could prove false statements about the natural numbers but still be consistent. I made this argument here. In contrast to consistency, soundness is not generally to be expected; in order for a collection of axioms to be set-theoretically sound, every one of them must be true. If the axioms were chosen randomly this would be highly unlikely. So the question becomes "what grounds do we have for thinking that ZFC is not merely consistent, but also (to keep things simple) arithmetically sound?"
I personally don't think we have strong reasons to think ZFC is arithmetically sound. Although a (purported) philosophical justification was developed later, when Zermelo first presented his axioms his justification was wholly, and explicitly, pragmatic. They weren't chosen because he thought they were "true". Maybe the thing that bothers me the most is that virtually all of normal, mainstream mathematics can be formalized in much weaker, essentially number-theoretic systems that do have a clear philosophical justification. What ZFC adds to this is a raft of set-theoretic pathology. To me, that suggests (but only suggests) that ZFC might not be arithmetically sound.
A: I interpret the question factually: Do there exist professional mathematicians who seriously doubt the consistency of ZFC? The answer is yes. Here are two examples (though sadly, both mathematicians in question passed away not too long ago).
Vladimir Voevodsky has said, regarding the possibility of an inconsistency in first-order Peano arithmetic, that "I'm quite seriously suspecting that such an inconsistency can at some point be found."
Edward Nelson at one point seriously believed that he had proved that primitive recursive arithmetic is inconsistent. Though he acknowledged his error when it was pointed out, he continued to believe that first-order Peano arithmetic was probably inconsistent, and certainly that its consistency was an open question.
Nik Weaver has written a number of essays, such as Mathematical conceptualism, in which he appears to cast doubt on the consistency of ZF.  But since he is an active member of MathOverflow, I will not put words in his mouth, but will allow him to state his own opinion on the matter, if he so chooses. (Deleted because Nik Weaver has already responded.)
A: George Boolos, eminent philosopher and logician, wrote as follows (perhaps slightly tongue-in-cheek):

He continued for a bit, and then...

(From G. Boolos, Logic, Logic, and Logic, Chapter 8: "Must we Believe in Set Theory?")
So, there you are: the answer to your question is "yes". At least he suggests that perhaps "set theory is not true" (not the same as "formally inconsistent"). The cardinal he mentions is not normally regarded as "large": it isn't even weakly inaccessible. The proof that it exists is easy.
My personal knowledge of formalising mathematics in the proof assistant Isabelle/HOL suggests that ZFC is much, much stronger than necessary for most mathematics with the obvious exception of the study of ZFC itself. A great body of advanced mathematical constructions – even Grothendieck schemes, thought by some to require more than ZFC – went easily into Isabelle/HOL. Higher-order logic is weaker even than Zermelo set theory, which (lacking the axiom of Replacement) is itself much weaker than ZFC. So if ZFC were somehow found to be inconsistent, mathematics would survive.
A: 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.

