The classic example of the Coase theorem (it is the classic example because Coase brought it up) is what happens when sparks from railroads set fire to farmers' crops. The efficient solution to this problem is either the railroads compensate the farmers for moving their crops away from the tracks or the farmers pay the railroads to put equipment on their trains that eliminates the sparks that cause the fires.
Climate change offers a better example. Today's LA Times had a story about the current conference in Bali that suggests that nations should be paid to be green. This is a perfect example of Coase in action (this is a Chuck Norris moment for economic theory). By paying countries to not pollute, green standards are met and everyone is better off. The resulting contracts ensure an efficient outcome where rich countries can continue to live as they wish, they simply have to pay for it. The poor countries are compensated for not being able to develop polluting (and likely low cost) industries. This is a better example of the Coase theorem, if only because it resonates with people more than railroads, than his original example.