In Present-day US English you may go for years—I mean that quite literally—without hearing shan’t. The only people likely to say it are those with a taste for pre-WWII British literature who have picked it up from their reading.
Note, though, that you may also go for years without hearing anybody say I shall. This use was already defunct when I was a child in the 1950s, despite the efforts of schoolteachers to require shall in place of will in the first person. Today shall is reserved, even in formal writing, for
pronouncements of a legal or quasi-legal character; it no longer signifies ordinary prediction of the future, but future requirement:
The party of the first part shall make an accounting quarterly to the party of the second part.
rhetorical assertion of unshakeable determination:
They shall not pass.
I shall return.
ADDED at Peter Shor's suggestion: It's sometimes used in first-person questions as a polite suggestion: Shall we go? or Shall we postpone that til we know more?
So if your interlocutors employ shall, they have no cause for complaint if you call them out by responding with shan’t; and if they are puzzled, it will give you an opportunity to demonstrate your superior mastery of the English tongue by explaining, condescendingly, what I have set forth above.