Nashville has started a bicycle program called Nashville Bike Share to encourage people to ride bicycles rather than drive. My first thought upon hearing this was "This will be the Berry Bikes disaster all over again". In 1998 Berry College's student government placed bicycles around campus; anyone who wanted to use one could hop on and ride it to class for zero price. As detailed by Frank Stephenson and Daniel Alban in this Freeman article, it didn't turn out very well. Riders treated the bicycles poorly, and within weeks they were unusable. Why? They were common property; because everyone owned them, no one owned them, and because no one owned them, no one had an incentive to take care of them. No individual bore a significant portion of the costs of damage to a bicycle. This is the classic Tragedy of the Commons. The program was eventually abandoned.
I don't think the Nashville Bike Share will end up faring quite so poorly. Riders must check out the bicycles and return them to the checkout one hour before the checkout location closes, and hefty late fees are assessed if riders fail to do so. I don't see anything on the site about fees for damage to the bicycles, but with the riders being individually identifiable, simple legal remedies should work. It will ultimately come down to how diligent the bike share workers are in chasing down bicycle abusers. This is a huge improvement on the Berry Bikes.
Does this mean the program is a good idea? It's hard to say. On the one hand, if there is sufficient demand for bicycles for rent, then it should be profitable for someone to provide those bicycles. The fact that no one is doing so suggests that this is not a wise use of government funds (I'm guessing it's a government program--it doesn't actually say so on the site, so I'm not sure, and in any case, it could also be a poor use of private charitable funds). On the other hand, if there are unpriced negative externalities from non-bicycle transportation (air pollution and congestion from cars, for example) then more bicycle use could be efficient. If that is the case, however, then it makes more sense to raise the gasoline tax, since that allows individuals to find the lowest-cost ways of reducing externalities.