Same as anywhere else, you can rent or buy. A few years ago buying was a fantastic deal, as house prices had not risen to match Danish earnings or house price booms in other parts of Europe, so there has been a major catch-up period with prices rising approximately 25% per year. This boom has been further fueled by low interest rates and financial engineering products such as interest-only loans. Perhaps the bargain in buying is less so now, although the fact that the national newspapers periodically print "house price crash" stories reminds me of Britain circa 2000, so maybe there is still some way to go. It depends on where you work - the big cities will have rental and purchase prices similar to other big cities, but if you happen to be working in a smaller town on Jylland, for example, you can still see decent houses on large plots of land for under 1 million krone.
By the way, you can
only buy a house if you're fully resident here. I think that they worry rich Germans will pop across the border and buy up all the houses as holiday homes, which would then be a problem collecting taxes from people not resident here, so somehow Denmark got special EU exemption.
As for renting, I
cannot speak from any personal experience, but I do know that our local newspaper has a full page of rental accomodation each week and I've no reason to suspect any different elsewhere. This search engine should prove useful.