This is such a tough question to answer as has been said! The rarest is one we don't know about, naturally.
I think the rarest American axe pattern is probably, maybe, the Hoosier. Though there are other patterns I have never seen an example of, so the statement seems empty. See? Lol.
I'll share a few of the rarest axes I own.
I attached two Hoosiers. Hopefully you can view them. First post using the new format.
Also you'll see a very clean Fulton Clipper etched axe. Made by Kelly, almost certainly in Alexandria, Indiana.
And as was mentioned above a 2-1/2lb Jersey pattern. Lugged boys axes are very uncommon.
View attachment 1575906
However, this 4lb Isaiah Blood and its 1-1/4lb counterpart feel like the rarest axes I own. They were made
In the mid 1800's.
I realize you didn't ask;
"What are the rarest axes you own"?
But I thought these would illustrate how even the "rarest" of axes can be in someone's collection, thereby disqualifying them from being the rarest axes. Ha.
I think hands down the rarest category are the new old stock axes. Paper labels, original paint, original haft and factory hang (if it's hung) etc. From the early 20th century and prior.
Similar to this Warren Axe & Tool Co fire axe;
Now I'm remembering I should've shared my pioneer by Emerson Stevens.
Not nos by any means, but only 1 of 2 known to exist;
Okay I'll stop now. Haha.
So I suppose none of these can be considered the rarest, but they are rare indeed.