According to an article by Yoon Ja-young in the Korea Times, there is a big push underway in South Korea to get accident insurance coverage for bicycle riders.
The article states, "Cyclists have been marginalized from insurance coverage.... Bicycles are categorized as cars according to traffic law, and are subject to the same regulations as cars in case of accidents. If a cyclist collides with a pedestrian on a sidewalk or bicycle path, for example, the cyclist is liable. Cyclists might face criminal charges as they are not insured. Consequently, an increasing number of cyclists have been calling for insurance products."
There are many problems with insuring cyclists. "There is also skepticism whether bicycle insurance would work. Insurers may be reluctant to sell the policies due to a relatively high loss ratio. Samsung Fire and Marine Insurance, for example, launched a product for bicycle riders in 1997, but suspended it after a few years due to growing insurance money payouts... Though bicycles are more vulnerable to accidents, ascertaining who is responsible for an accident isn't always clear. Since bicycles are not registered, the potential for fraudulent claims is high."
The thought of "cycling insurance" leaves me feeling conflicted. On the one hand, it would be nice to be covered for damage I might do in an accident. On the other hand, one of the wonderful things about riding is getting away from the whole automotive bureaucracy that squeezes so much cash out of so many people. Furthermore, I strongly suspect that if cyclists CAN buy accident insurance in the USA, it will not be long before some lobbyist pushes for a law that requires cyclists to do so.