Københavns Museum

When I visited in August 2014, the museum was a mix of Copenhagen history, impenetrable Kierkegaard stuff and the local environment. I think Americans will particularly appreciate the immigration stories in the Copenhagen history section (we don’t realize how much the rest of the world has been affected by immigration now and historically) and planners and geographers will appreciate the environment exhibition, which touches on climate change.

Great museum for English speakers as all the exhibits had English text or title language.
I went for the current Kierkegaard exhibit in celebration of the 200^th anniversary of Kierkegaard’s birth — any Kierkegaard fan in the time zone or really, on the continent, should make the trip. The exhibit included many of Kierkegaard’s personal affects: writing desk, couch, curio (where he kept items reminding him of Regine), a lock of his hair, and the returned engagement ring from Regine. Plus, excellent narrative and fitting Kierkegaard excerpts throughout.
The exhibit is intimate, well-​designed and beautiful. For a Kierkegaard fan, it’s a once in a lifetime opportunity and the museum has done a wonderful job celebrating one of the greatest minds of all time.

This was my favorite museum out of my trip to Copenhagen. It has excellent English explanations for most of the exhibits which is so important to understanding what you’re looking at; many of the other museums did not have good or any English explanations.
They had a very cool exhibit called «What it means to be a Copenhager» which tells the stories of several people who moved to Copenhagen. It gives a really personal touch to the lives of these immigrants. It was also nice to get a sense of what their daily lives were like and for me to learn more about Copenhagen.
There are several interactive exhibits which are also available in English. Overall, this is a great place to visit.
Tip: Admission is free on Fridays.