Copenhagen  

I second
everything I said the last time someone asked this (link below).
That said, I'd also add that if you want to stay in a relatively inexpensive--but interesting--place, check out the Cab-Inn. Their rooms are tiny but so well designed and laid out that you never feel cramped up. Well, I didn't, anyway--and the first time I stepped foot into one, my heart sank at the prospect of sharing such a tiny space for 5 days with my husband. But it was not at all what I'd imagined it would be! Chairs and table fold up and hang on the wall, beds double as sofas, and there are open shelves for your (folded) clothes and stuff. It took a bit of doing to figure out how to shower without getting everything wet (put towel and toilet paper on high shelf over toilet--duh).
Also--big, wonderful breakfast spread every morning for about $10, all you can eat bread, rolls, hard- and soft-boiled eggs, 2-3 cheeses, cured meats, granola, yogurt, pastries, juices, tea & coffee, and some commercial cereals, etc.--all enough to keep you going for hours.
I've stayed there many times since and would recommend it to anyone. It's comfortable and convenient and I would rather save my money to spend on stuff to take home.
For other fun, check out the flea markets on Saturday mornings starting in...late May? Mid May? Not sure. There is one over at the town hall in Fredriksburg, which is maybe a 10-15 minute cab ride from Copenhagen or a 20 minute walk from the Cab-Inn on Vodruffsvej, just across the lakes from Copenhagen (though not a very interesting walk, and not real direct). I got some beautiful Danish pottery pieces for under $15 each, and one of those sleek Danish design desk phones (flat, rectangular) for a few bucks from a dealer who had a bunch of them. That was 5-6 years ago, dunno if it's changed much since then.
http://www.designaddict.com/design_addict/forums/index.cfm/fuseaction/th...

+1
for Illums Bolighus. I could stay in there all day. There's also a little design-oriented shop directly across the street from the entrance to Tivoli Gardens, (I forget the name), which has lots of Marimekko, Alessi, etc.
We stay at the Radisson Blu Royal Copenhagen simply to be surrounded by all of Arne Jacobsen's magic.

Hi Luna. Are you looking for...
Hi Luna. Are you looking for new or vintage design?

luna
One thing that I didn't know when I first went there was that pretty much every store sticks to the suggested retail price for stuff, so if you see something you like, just get it. You're not going to see it at a better price somewhere else. (Obviously this goes for new stuff only!)

Thanks for all your help...
Thanks for all your help guys and gals. Good tips, I'm really looking forward to it. Riki, I think the Rad Blu will have to wait until I strike it rich 🙂
Spanky, I like the sound of yours but I'm a little claustrophobic.
M_Andersen, mostly vintage, but also some new design small pieces that are hard to get elsewhere.
I plan to go to Louisianna to pick up a few books (hopefully!)
Any ideas for good flea markets?

flea market
This is the one that I have been to a few times and it's where I got the pottery mentioned above. I see they're now open on Fridays too---dunno if that's a newer development or if I just missed it before. I would go on Friday rather than Saturday.
I will definitely try to hit this when I go again. I got good stuff for cheap.
http://www.visitcopenhagen.com/Events/Copenhagens-Flea-Market-at-Israels...

great link, thanks spanky!
great link, thanks spanky!

Ok, Luna. I know just about...
Ok, Luna. I know just about every store that deals with design in Copenhagen, both new and vintage. I will gather the names/adresses of those worth a visit and post them in this thread within a short time. Regarding fleamarkets there's usually a lot going on but it depends on when you are coming to Denmark?

Living north of Copenhagen...
Living north of Copenhagen I've never stayed at a hotel in the city, however these three hotels/hotels are located in the city center and are especially for people who like design.
http://www.hotelfox.dk/
http://www.danhostelcopenhagencity.dk/cphhostelscom/
http://www.hotelalexandra.dk/

Sod all the stuff
Above and get yourself down to nyhavn færgekro and sample their smorgasbord lunch with lots of snaps.
Once the snaps hits the system you wont care about shopping but just chilling.
Alternative lookmat the markedskalender which is a list of all carboot sales in denmark
http://Www.markedskalenderen.dk

Genbrug
Keep an eye peeled for GENBRUG signs (same as Thrift Shop). They all seem to keep odd hours and are only open a few days a week, but you can find some nice things if you are lucky.