I've seen pretty much every AKC breed and then some. That's what happens if you end up at enough dog shows, lol.
I see things like Canaan dogs a few times a month. Have met roughly 15 pyrsheps, 5 Cirneco dell'etna, Danish Swedish Farmdog, several mudi/croatians, lots of pumi, etc. Welsh sheepdogs are extremely rare in the US but I have met 5 that do dog sports around here. My trainer has one. My other trainer has a lacy, which aren't super common though I saw them a lot in Texas.
I guess the rarest would be some New Guinea Singing Dogs and I didn't even see them at a dog event!
I've met a Petit Griffon Basset Vendeen, a Spinone Italiano, a Greater Swiss Mountain Dog, a Leonberger, and some Kelpies (maybe not rare, but uncommon here!). We also have a Potcake that comes into work but I think that's probably more of a type than a breed. I've also owned, and known a bunch of other Swedish Vallhunds.
Not counting dog shows, I've met a briard and spinoni. I know LostandConfused's English Shepherd, Hudson. I know a basenji who does nosework, and two Catahoula leopard dogs (rare up here anyway). Welshies are rare and I've met a bunch of those. That's about it. Everybody else around here seems to have labs, GSDs, huskies, or mutts with a few herding or sporting dogs thrown in.
I'm not sure what counts as rare either, around my neighborhood it's all labs, goldens, pit bulls, and hound types (and mutts) so pretty much everything is rare to me. At work we have a Basenji, we also have a blue lacy. I don't know if the owner knows she's a blue lacy (she's not listed as one but that might be because it's not in the computer), but I have little doubt. One of the people I work with has a Tibetan Mastiff that comes in occasionally. Not sure what else, nothing super out there.
Hmmmm... I'd probably say Irish Wolfhound. There's one who lives in my general neighbourhood and it's probably the only one in town. I know one or two whippets in the town as well. There are a handful of other greyhounds. I had a Munsterlander come in recently. Don't think I've ever met one of those before.