Maybe surprisingly there is an airport in Hanstholm with two daily connections to Copenhagen. The price is even very reasonable. You can get a stand-by ticket for 410 kr. one-way from/to Copenhagen. Recently the route has become a bit more popular, so it seems that you now risk not getting on with a stand-by ticket, especially Monday mornings. Stand-by tickets can only be signed up for 14 days in advance.
Tickets can only be bought on their website (sorry no English) but I try the phone number below if you get stuck. No Amadeus booking.
However there's one major drawback. They cater mainly to business people, so they don’t fly weekends and not during the summer holidays!!
The trip itself is an experience for people used to only flying jumbo jets. The plane is a small two engine propeller aircraft with 15 seats and the service is a picnic basket passed down the aisle when the plane is airborne. The plane is too small for anybody to stand up, and there’s no stewardess anyway, just the two pilots. I thought it was great fun and it takes only roughly 50 minutes as opposed to the 5-6 hours by train.
There is a new ferry company called Master Ferries starting a route from Hanstholm to Kristiansand in April. The trip will be less than 2 hours as they have bought the old Mols Line ferry previously sailing between Zealand and Jutland.
As an opening offer it will be possible to buy a one way ticket for 9 to 79 dkk. Tickets go on sale in January, but I have yet not been able to find their website.
From Hanstholm you can take the ferry to Egesund, Haugesund and Bergen on the Norwegian west coast with the newly build Fjord Line ferry.
Updated with link to their brand new English website.