The Danish government is going to recommend that we buy 27 F-35 Lightning II to replace our aging fleet of 42 F-16

.
The recommendation isn't final but in the end it is most probably a question of a few more or a few less aircraft than the proposed 27.
The final candidates were the F/A-18F Super Hornet, Eurofighter Typhoon and the JSF. Saab withdrew with the Gripen E/F beforehand because they figured that they didn't have a chance.
In the public debate the Super Hornet has been portrayed as the sensible, cheap and reliable choise that will do the job (currently bombing brown people and intercept the russians on occasion) well enough. But it is a little long in the tooth already and the availability of updates throughout it's lifespan is questionable.
The Eurofighter is a very capable dog fighter and is maturing as a strike fighter. It should be relatively reliable since it has been in service for a decade now. The costs aren't that much lower than with the JSF though - on paper at least. Denmark has also been very eager to be best buddies with the US so politically it doesn't really fit the profile (sadly).
And then we have the JSF. I guess I don't have to list the issues it is facing. Denmark has contributed about $ 230 mil. USD to the development of it as it stands and we really want some brownie points with the US so it was the expected outcome. But many worry that the costs will run wild and that the plane will never realise all the potential that the experts say it will some day have. It also looks like the current costs of the purchase, before any budget overruns, will be a severe burden on all our armed forces, meaning budget cuts across the board.