Button Men is a free Print&Play game for 2 players. Each player chooses a character card, and the polyhedral dice associated with that character, and takes turns trying to capture their opponent's dice. There are over 200 characters created for it. Dice with more sides can capture opponent's dice more easily, but they are worth more to your opponents when they are captured. Some analysis has been done with regard to strategy, but have the clearly best and worst character cards emerged?
There was a "Button Men Online" website until recently that had amassed HUGE quantities of statistics on different Button Men competing (albeit being controlled by people users on the site). If you could get a hold of that data, that might answer your questions.
I've messaged @cheapassjames on twitter, he might have the data. Possibly, you could contact the site creator, Dana Huyler (but the email email@example.com seems to be out of order).
(disclaimer: i haven't played button men seriously in ages, but i'll share my impressions from back in the day)
My feeling is that Lab Rats is the best button, as its 2-dice will allow skill-attacks on far better dice than it has to give away. Bunnies, with its 1-dice is also strong, but can too-often get trapped into not being able to make any attacks.
The worst by far has to be the dragon with d30s all down the line. Yes, it can take anyone's dice, but odds are you're going to roll one of them low, and give up 30 points at once. Do that twice and there's no way to win against most buttons.
Some are banned from tournaments for power. i think lab rats and bunnies were both banned at some point. Some are banned due to non-playability (how would you roll a pi-sided die for the James Earnest button?), and some are banned for their swingy nature (the xxxenophile buttons, with their 3 x-dice, were considered too "swingy"; you could completely change the nature of your button between rounds, though it looks like they're back on the permitted list) http://www.beatpeopleup.com/tourname.html