Picking a top 20 or so places in Copenhagen wasn't hard a decade ago. Happily it's now a much tougher prospect - the dining scene has improved vastly and the options are far greater both in terms of places and cuisines.
My criteria are first and foremost that I look for kitchens that use fresh ingredients to make dishes themselves as fresh as possible. Having reviewed restaurants from the high to the low end all over the world for a variety of guidebooks, it's something I soon come to crave on the road. I've also been an assistant chef and washer upper in a few places in my youth and I know the difference between kitchens that work hard to create flavourful dishes and those that rely on wholesale caterers and whose only contact with food is during its brief transit from the freezer to the fryer.
This doesn't mean I always pick pricier or fancy-pants places with wine waiters and elaborate menus, although I do think you tend to get what you pay for. There's nothing wrong with down to earth food, in fact it's often my favourite option. You can usually smell a good restaurant as soon as you open the door. Fresh herbs, or fresh bread or perhaps a whiff of the cheese trolley are always things that get my juices going.
This is an expensive city and there are quite a few high end places on the list. My reasoning is that you've come to the Michelin star centre of Scandinavia and the home to the world best restuaurant so why not sample some of the best? That said there is, I hope, a decent selection of good honest places offering value for money and bags of flavour to go with it.