This is by no means comprehensive, but are where I’d go:
Noma. It’s no El Bulli, but this is open, so! I love how it looks like a nondescript warehouse from the outside.
The Opera House. Ridiculous inside. (It looks like a pumpkin.) The Royal Danish Ballet and Royal Danish Orchestra both perform here all the time and apparently THE QUEEN comes all the time.
Koncerthuset. It’s a theater with 4 semi-transparent screens for walls. So you can see inside during the day and watch projections on it at night.