Copenhagen is a very low city in terms of building heights thanks to age old building laws so the novelty of getting up high above the city is popular with locals and tourists alike. You can really get a different perspective of the city and its fun to try and spot your home or the roof tops of familiar buildings.
Here are my favourite spots to see the city from above, enjoy a drink or a meal or just watch the sunset.
Bella Sky Bar at the Bella Sky Hotel
Sky bars are increasingly popular in major cities and the relatively new Bella Sky Hotel in Ørestad offers a luxury chance to see the skyline of Copenhagen from a distance in one direction, the new architecture of Ørestad in one and open space and Sweden in another. Enjoying a glass of bubbly and watch the sunset from here is a breathtaking experience. You can also enjoy a smørrebrød lunch here.
The Round Tower is probably one of the best know spots to see the city from above and after over seven years of living here, it is still my favourite. I love to gaze over the red roofs of the old city and further out to the sea. It is also great to spot places in the other areas of the city and I love to try and spot my own apartment. The tower is worth a visit any time of the year as you can really get a feel of the city around you from here.
Tårnet at Christiansborg
This is a new kid on the block in terms of public access. The tower at Christiansborg (Danish Parliament) has been open to the public for less than a year and gives a higher view than the Round Tower and a different perspective on a similar view. There is also a restaurant at the top of the tower, perfect for a romantic dinner a deux.
Sticks & Sushi at the Tivoli Hotel
Like the Bella Sky Bar this is another fabulous place to watch the sunset. The restaurant is arrange to give diners the maximum views of the harbour, Tivoli and the new developments in Sydhavnen. On a summer evening make sure you take a table outside to get the maximum enjoyment of the sunset. Inside there are swing seats facing a huge glass wall, affording fantastic views of the harbour, perfect for enjoying an after dinner drink. Just make sure you book a table as this place gets busy
Copenhagen Zoo is situated on the highest spot in Copenhagen and a short trip up the iconic black and white tower gives amazing views over the city and to the Øresund Bridge, and also out towards the countryside. Don’t miss out on this when you are visiting the zoo.
Dejlige Days 
I would add the Vor Frelsers Kirkes Tårn in Christianshavn (http://www.vorfrelserskirke.dk/taarnet) to the list. The views from there are spectacular!