By the standards of other fields, mathematics doesn't have "substantial" controversies, but here are a few examples, including the two mentioned by Morendil and Jonathan L.

1. A century ago there was controversy about constructive mathematics and about axioms like choice. I think these controversies were resolved by driving out the people with high standards into logic and theoretical CS (Informatics). The higher standards seem more useful in those places, but to call the current standards "uncontroversial" is, I think, misleading. Mathematicians claim to accept that constructive proofs or proofs in weaker systems are in some sense better, but in practice they never think it worth the loss of elegance. In recent decades, the desire to computerize calculation has increased the popularity of constructive proof some. For example, the original controversial non-constructive proof by Hilbert of his Basis Theorem (1890) is usually taught following Noether's very short proof (1921) rather than using Buchberger's constructive Gröbner bases (1965).

1b. Bourbaki and especially Grothendieck actually work in stronger systems than ZFC. It is not clear to me how controversial this is, but there have been awkward conversations (eg, on the FOM mailing list) about the status of Grothendieck's results.

2. Mathematicians have a strong consensus about the ideal of proof to which they aspire: formal ZFC, but they don't actually produce formal proofs. The real standard is informal proofs and a belief that they can be debugged to a formal proof. This is not a precise standard. People are generally nervous about very long proofs. They are not very nervous about the thousands of pages of SGA that go into Deligne's proof of the Ramanujan conjecture because most of that is modular. It breaks into small pieces which are easy to understand and which are reused in many places, thus extensively tested. But the proof of the classification of finite simple groups (famously condemned by Serre around 1980) is made up huge modules that are used for nothing else. In fact, it contained an error at a high level, that the organizers failed to observe that one of the assigned modules had not been delivered. There will be more confidence in the second generation proof because (1) it is projected to be only 5000 pages and (2) the same people will prove the modules as will use them.

2b. When someone rewrites a proof in his own words, it is much more reassuring than if he simply claims to have read and understood it. Modifying the proof, either for simplification or to prove something else is also helpful. But it is generally rude to express such doubts publicly. Here is someone getting in trouble for voicing such concerns. There are lots of decades old results that are controversial in some circles.

2c. It is quite common for top mathematicians to provide fewer details of proofs. Often other people write longer papers filling in the details. The status of the result before this is sometimes controversial, but again often not publicly.

3. The computer-generated proofs of the four-color theorem are controversial for two different reasons. One reason is that since they don't exist in a human mind, they don't provide insight. The other was that since they were of a different form than usual proofs, they were not trustworthy. In fact, there have been many generations of these proofs and I am told that each new one claims that the previous one missed some cases. The latest one generated a formal proof, so it is computer-checked and as defensible as anything. But the first controversy remains.

4. Physicists often make claims about mathematics. For example, that the scaling limit of the self-avoiding walk on the square grid exists; and that the limit has a particular fractal dimension. Is this a controversy that the mathematicians do not accept the physicists' "proof'? Or do they just mean different things by that word? Anyhow, the physicists go on to say that the limit is a conformal field theory, and the mathematicians reject this as a meaningless statement, though they try to salvage it, eg, through SLE.

5. People make calculation errors all the time. Long calculations can be controversial, but they probably should be trusted less than they are. It is generally better to compute many things in parallel, so that an error in the calculation infects all the answers and is immediately apparent.

Curiously, participants in discussions of all of these subjects seem equally confident, regardless of the field's distance from experimental acquisition of reliable knowledge.

As someone in academic sociology, this conflicts with my experience. Seeming confident is a matter of social skills, temperament, and the conditions of winning arguments with peers in the same area. But as best I can tell, in terms of assigning probabilities, sociologists can be divided into those who think the field (or at least the parts they're engaged in) to be scientific in the same manner as the physical sciences are scientific, but much harder and with greatly reduced ability to assure confidence in its findings, and a minority who think that society is so complex (especially when "meta" epistemological factors are brought in) that confidence in any of its conclusions beyond dumb facts is unwarranted and that science is a bad term for it. Claims people state with great confidence tend to be ones that enjoy broad agreement.

It would be very surprising if hard scientists were mostly mutants.

There are few in physics, chemistry, molecular biology, astronomy. There are some but they are not the bulk of any of these subjects.

How do you define the "bulk of the subject"? Take physics from wikipedia:

Before the 1990s, string theorists believed there were five distinct superstring theories: open type I, closed type I, closed type IIA, closed type IIB, and the two flavors of heterotic string theory (SO(32) and E8×E8).[13] The thinking was that out of these five candidate theories, only one was the actual correct theory of everything, and that theory was the one whose low energy limit, with ten spacetime dimensions compactified down to four, matched the physics observed in our world today."

Having five competing theories seems like controversy to me. It's just that as a layperson I don't understand what the core of the controversy is about.

Molecular biology is the only one of those where I took classes at university. I don't think that the field is without controversy. It's just that outsiders don't get much information about it.

The stated goal of the human project was: Project goals were to among others:

determine the sequences of the 3 billion chemical base pairs that make up human DNA,

They didn't determine all the basepairs of human DNA. After getting to 92% the declared their project a success. They then pretended that the other 8% doesn't matter. Happy coincidence isn't it? The stuff that's hard to sequence is also practically unimportant...

The extend to which those 8% matter is up for debate.

Determination of protein structures is a bit similar. Most of the protein sequences we have were created through crystellizing proteins. We don't really know exactly how big proteins move when they aren't crystallized and are surrounded by water and other stuff that's in the cell. It's practical to assume that we basically know how a protein looks like when we know how it looks like when it's crystellized.

For outsiders it's easy to understand history, psychology or sociology controversies. For the hard sciences it's hard for outsiders to understand the controversies within the field.

I think controversy is more closely related to the number of people who feel qualified to participate in discussions. The hard sciences exclude idiots by the very language they use. In softer subjects like sociology, or barely academic ones like politics, anyone can pick up the standard textbooks of the field, bullshit his or her way into a passing grade and then continue to muddy the waters until retirement.

While I don't think it changes the order of the fields, I can think of three versions of "how controversial is X" that might yield different answers: a person from field X might trust all its results and say it is uncontroversial; a person from field Y might say it is uncontroversially false; clearly there is controversy if we put the two people in a room.

Even at the extreme, physicists make lots of claims about mathematics that the mathematicians reject. Sometimes the claims are precise enough that the mathematicians accept them as conjectures (and aren't bothered that the physicists use the word "proof"), but often the mathematicians cannot even tell what the conjecture means and reject the label "mathematics." I gave some examples here.