As of April 2008, a new anti-smoking law means that it is now illegal to smoke inside most places in Denmark - there is an exception for extremely small pubs, but those aren't usually tourist-oriented anyway. Trains are smoke free, the central train is as well (smoking is still allowed on the semi-outdoor platforms) and people are generally getting better at showing consideration. Street cafes have smoking areas outside on the street, where you can sit if the weather's up to it. 

The amount of smokers in Denmark is very close to the European average of about 22%, since there are so relatively few areas where people can smoke, expect these to be badly ventilated and full of tobacco smoke.