What would happen if ZFC were found to be inconsistent?
Solution 1:
The way I see it there are two possible outcomes:
The contradiction will come from a very concrete and identifiable part of the system, e.g. power set axiom and some replacement, which we can remove and replace by milder variants -- much like we did with Frege's comprehension schema. We will then go through the proofs and verify which of the proofs cannot be modified, and hopefully most of the big proofs will go through, in which case there will be some change, but not too much.
The contradiction will come from a much weaker theory, e.g. Peano Axioms or some other mathematically useful theory. In this case it's more than set theory which has a problem. It would mean that many people will now begin to look for alternatives, possibly in several fields. I suppose that set theorists will also try to find themselves an improved theory.
In either cases there will be a lot of work to see which fragments of ZFC we can rescue, and how many of the proofs would go through. Note that Russell's paradox, at the time, wasn't considered as a big deal by non-logicians. So while the second option would affect many mathematicians, the first option would be largely ignored by most (or it will be used as a source to kick set theory around).
If at some point we conclude that we cannot rescue a lot of the reasonable proofs (e.g. some inner model theory, forcing, etc.) then it is possible that ZFC will be just dropped and a new basis will be sought.
Of course it is possible that just something about large cardinals would be inconsistent, and then we will be able to draw a lot of conclusions about the universe of sets (for example, all propositions which imply the consistency of this cardinal would be immediately false. If you have found a contradiction in a measurable cardinal then you can prove that AD is inconsistent with ZF). But that's still quite far from showing that ZFC itself is inconsistent.
Solution 2:
There was a quite relevant post at here:
https://mathoverflow.net/questions/40920/what-if-current-foundations-of-mathematics-are-inconsistent-closed
and the original lecture is worth watching.
Solution 3:
I'm personally a fan of ETCS. See, for example, Todd Trimble's blog post on the subject.
Solution 4:
Note that an inconsistency implies that any theorem and its negation can be proven. The fact such a thing hasn't happened (no one has ever proven the negation of a theorem in mainstream math) means that wherever ZFC fails, it is almost certain that a more restricted set of axioms can be found where most math can still be done.
To see an example of what could happen you have to look no farther than Russell's Paradox. It shows that naive set theory makes no sense, but the example is far removed from the regular sets (with "actual elements") that we all usually use, and axioms where set theory works were found.