Thanks to this thread I was able to make it to just about all of the stores that stocked anything I'd be interested in. I figure I should report on what they stocked just in case someone else is travelling there soon. I'll refer to Oslo/Stockholm too. I assume most visitors would be heading to one of those cities as well. This'll be limited to menswear. I didn't look at the ladies' stuff.
Copenhagen
I ended up having the most luck at Birger Christensen. They had some nice Jil Sander, Rykiel Homme, Margiela (10 + 14 I think) and Prada pieces. No men's McQueen or Miu Miu. I never found anything from either line. The SA's were helpful and perceptive. One guy directed me to Jil Sander before I asked where to find it.
Strangely, the smallest Jil dress shirts were 39. The stretchier ones almost fit me, but if I'm paying full price I'm not going to get something one size too big. They did have smalls and mediums for the tees and such. I think they had Dolce, Galliano and DSquared2, but I didn't look closely. I saw some Paul Smith socks too. I'm not sure if they had his full line.
As Baizilla said, 1206 was definitely worth a stop. His description of their stock was accurate. They also had Acne and Rykiel. It's where I found my Kris Van Assche shirt. Out of town visitors should note that they only accept Danish credit cards. There are ATM/Bancomats close by, so it's not a major issue, but it's worth noting.
Storm seemed nice, but it was the last of the shops I visited, so I didn't end up getting anything. They had Dior Homme, Burberry Prorsum, Dries and a few other things that are slipping my mind.
I never went to Burberry, so I don't know if they stock Prorsum in the boutique.
Oslo
I didn't have much luck here. The only interesting store I found was Kamikaze. It was pretty much your typical luxury brands- Dior, YSL, Prada, Gucci, Dolce and Gabanna. Apparently they never put anything on sale. The man tried to sell me some S/S DH stuff at full price. It was disappointing. With a name like Kamikaze I was expecting something Japanese!
Stockholm
From what I could gather, Stockholm only had two shops that stocked interesting menswear. NH things in a couple of different areas. What I guess was supposed to be the edgier, more youthful area had Marc by Marc (and maybe the full line), V&R, Y3 and Acne. I was really only interested in Rolf and the selection wasn't too great. There was another area with Jil Sander and some other more classic lines. When I was there they had ties, tees, a pullover (which I bought) and a suit, but the SA mentioned that was just the preview. They were supposed to get a bunch more about a week ago: white shirts, more suits, coats.
I can't remember the name of the other shop, but it was around the corner from the Gucci boutique. It was a woman's name. They had YSL, Prada, Burberry, Botega shoes and some Lucien Pillat-Finet sweaters. They didn't seem as friendly. I gave them my standard "Do you stock....?" and "Does anyone else in town?" and the SA basically said there wasn't another shop in town that was close to their level. This is a far cry from the guys in BC and 1206 in Copenhagen. They were glad to tell me where else to browse.
Hopefully this'll help some future travellers.
Copenhagen
I ended up having the most luck at Birger Christensen. They had some nice Jil Sander, Rykiel Homme, Margiela (10 + 14 I think) and Prada pieces. No men's McQueen or Miu Miu. I never found anything from either line. The SA's were helpful and perceptive. One guy directed me to Jil Sander before I asked where to find it.

As Baizilla said, 1206 was definitely worth a stop. His description of their stock was accurate. They also had Acne and Rykiel. It's where I found my Kris Van Assche shirt. Out of town visitors should note that they only accept Danish credit cards. There are ATM/Bancomats close by, so it's not a major issue, but it's worth noting.
Storm seemed nice, but it was the last of the shops I visited, so I didn't end up getting anything. They had Dior Homme, Burberry Prorsum, Dries and a few other things that are slipping my mind.
I never went to Burberry, so I don't know if they stock Prorsum in the boutique.
Oslo
I didn't have much luck here. The only interesting store I found was Kamikaze. It was pretty much your typical luxury brands- Dior, YSL, Prada, Gucci, Dolce and Gabanna. Apparently they never put anything on sale. The man tried to sell me some S/S DH stuff at full price. It was disappointing. With a name like Kamikaze I was expecting something Japanese!
Stockholm
From what I could gather, Stockholm only had two shops that stocked interesting menswear. NH things in a couple of different areas. What I guess was supposed to be the edgier, more youthful area had Marc by Marc (and maybe the full line), V&R, Y3 and Acne. I was really only interested in Rolf and the selection wasn't too great. There was another area with Jil Sander and some other more classic lines. When I was there they had ties, tees, a pullover (which I bought) and a suit, but the SA mentioned that was just the preview. They were supposed to get a bunch more about a week ago: white shirts, more suits, coats.
I can't remember the name of the other shop, but it was around the corner from the Gucci boutique. It was a woman's name. They had YSL, Prada, Burberry, Botega shoes and some Lucien Pillat-Finet sweaters. They didn't seem as friendly. I gave them my standard "Do you stock....?" and "Does anyone else in town?" and the SA basically said there wasn't another shop in town that was close to their level. This is a far cry from the guys in BC and 1206 in Copenhagen. They were glad to tell me where else to browse.
Hopefully this'll help some future travellers.