Out of curiosity, I did a google. One that seems to top some people's list of 'most rare' is the future perfect continuous passive. The example one writer uses is: “By March next year furniture will have been being made in this workshop for 100 years.” I think this tense you'd only consider for rarity in writing. In conversation probably 0% chance of hearing it being used. In actual conversation, I would guess the subjunctive mood is the least used. But that's purely my guess :)

I'm sure it must depend from place to place in native English speaking countries. You'd have to localize this question to small communities and then try to find and answer for all of them. Even them it might be futile since every person speaks differently so, in the end, you might even have to ask every single individual person.