I've had two that I would consider rare or unusual; one rare, one unusual.
The rare one: 2008 MV Agusta Brutale 910S. It lacked the pricey suspension, brake, and tuning upgrades of its 910R sibling, but it was...well...just a brutal bike, and only lacked about 6-8 hp as compared to the 910R. Absolutely brilliant on a smooth, twisting back road, with telepathic handling, fantastic brakes, and an engine that really came into its own north of 6000 rpm. It was an absolute wheelie machine...just think about twisting the right grip, and if you weren't leaned over the tank, the front wheel was coming up. WAY too easy to go flying past 100mph without realizing it. It hated being ridden in town...it would heat up fast (unless you swapped in a high-flow water pump, a better radiator fan, and a cooler thermostat), and fueling was snatchy below 4000 rpm (unless you installed a power commander and had it dyno tuned). I sold it for two reasons: due to a worsening back, I could only ride it for 10-15 minutes before pain set in, and because it lifted its front wheel so easily I knew it was going to kill me very, very quickly.
The unusual: a 2006 Hyosung Beaver 125 scooter that I owned while I was in Seoul. Not rare in Korea, but just the name made it unusual (for westerners anyway) and pretty funny. Typical comment from bystanders: "Hey Sir, nice beaver!" Or "Hey Sir, it's raining and you rode in today? You like wet beaver?" Yeah, yeah, laugh it up...you know you want to ride my beaver.
I sold it just before I left Korea in 2007.