Must-see in Copenhagen?  

The Museum of Danish Art &...
The Museum of Danish Art & Design is wonderful.
The Louisiana--MUST see. My favorite. Modern art and spectacular grounds looking out on the water. Fabulous sculpture gardens. Go when the weather is likely to be nice (though that's pretty hard to predict).
Go to Illums Bolighus department store (not to be confused with Illum, which is more like Macy's). Wonderful Danish designs that you can touch, feel, sit on, lay on, and maybe even buy if your pockets are deep enough.
The Frilandsmuseet (Open-Air Museum) isn't about modern design but it's a chance to see how Danes lived going back many centuries. I went there not expecting much but was entranced. Very, very interesting. They went all over Denmark for buildings dating to as far back as 1400 (?), dissasembled them and then rebuilt them on site in sort of a village. You can look into all of them and some you can step into to look around.
That's a start.
Oh, and be sure to eat some smørrebrød! And wienerbrød!
http://www.kunstindustrimuseet.dk/en/

Thanks!
Thanks, spanky* ~ All those suggestions sound great!
Still open to other thoughts, if anyone has them!

I've never been, but if your...
I've never been, but if your a Jacobsen fan, I've heard the Royal Hotel is worth a visit.

the Royal
Jesgord, I'm definitely planning on a trip to the Royal Copenhagen, as I am a big Jacobsen fan. Those are some great photos!

Hi squadrato. I live in Copen...
Hi squadrato. I live in Copenhagen and can help you with design-related links and adresses. Feel free to contact me at andersen1982 at hotmail.com

Finn Juhl's house, Hotel Alexandra
Definitely go out to Finn Juhl's house, which has been preserved as a museum a few miles outside the city center. It's technically closed for repairs but call and make a reservation, and they'll let you visit. Absolutely fantastic place.
Also, stay at the Hotel Alexandra if you can -- full of Danish modern gems and a lot of soul, and everyone there is a real design enthusiast. I stayed at the Royal this summer for a week and it's just a slick overpriced business hotel that uses Jacobsen as a gimmick -- drop by and look around the lobby (eggs and swans as far as the eye can see), and ask if you can have a look at the one remaining suite that is kept in Jacobsen's original design. The rest is all "updated" and pretty disappointing.
Also the Kunstindustri museum has a great collection of Danish modern classics.
Have fun!
http://www.ordrupgaard.dk/topics/collection-and-architecture/finn-juh l's-house/finn-juhl%92s-house.aspx

Copenhagen
Just booked a train, gonna spend four days there later this month.
Any suggestions on shopping / any further must-sees that have not yet been listed?
Best regards,
actionandaction

Hay House
Try Hay House, the HQ of Danish manufacturers Hay. It's a nice, eclectic space with the best of their designs showcased over multiple floors alongside other products by similar European manufacturers. Nice building too.
HAY HOUSE
OESTERGADE 61, 2ND-3RD FLOOR
DK - 1100 COPENHAGEN K
DENMARK

Hay web site
Forgot to include the firm's web site address.
http://www.hay.dk

loppies
If you like fleamarkets, small galleries and alternative distric...
http://www.ravnsborggade.dk/

Go Down 'bredgade'. There...
Go Down 'bredgade'. There you will find stores like www.klassik.dk and Carls Hansens (www.carlhansen.com) new flagstore. Take a taxi to Roxyklassik.dk - and if you fancy - go to the brand new www.denblaaplanet.dk
.....and remember the Little Mermaid 😉

Thanks
Thanks for sharing this