Walkin' the rabbit. It sounds like a cheesy dance, but うさんぽ really does mean to take a rabbit on a walk, leash and all. It comes from two words, as most of the slang I cover does: 兎 (usagi=rabbit) and 散歩 (sanpo=walk).
Now if there is a word for it, that means it must be at least a little bit of a popular phenomenon. In fact, I joined a mixi group, half out of amusement, that had the name うさんぽ and 440 members. Most of those people have a rabbit for their profile pic it seems.
I say half out of amusement because I am actually considering buying a bunny. I was exploring my town one day and came across a rather ghetto shop full of chickens and rabbits, and it struck me that not only is a rabbit a cute animal that won't claw my furniture, but I can feed it all those scraps and extra vegetables I end up with. Half of my carrots alone always go bad because I never get to them in time. I asked one of my privates about it, and she warned that they have lots of stinky poo, so I am not convinced I want to go through with it...
Speaking of cute pets, I had a J girlfriend who had a chiwawa. Her family pampered that thing and put it in a different outfit everyday. Much to their chagrin and mine, it loved me more than anyone in the family. Anyways, here is a pampered bunny usanpo video: