Thank you for your question. The answer is yes, as an American citizen you can apply for a temporary residence permit for Denmark, and then, after a number of years (depending on the purpose of your residence), this can be converted into permanent residence.
The options are quite large, and depend a lot on what you plan on doing in Denmark. To enumerate, there are residence permits for persons who: work (and here there are sub-options, such as Positive List - the jobs where they have a shortage of workers; or the Fast-Track Scheme - for highly qualified professionals); are entrepreneurs ("start-up Denmark"); do research; are herdsmen or farm managers; are looking for family reunification (if you have family members there); are artists, athletes or chefs; are pursuing a Danish Masters or PhD degree, with the aim of establishing permanently in Denmark. Also, you could obtain residence in Denmark in a roundabout way, by first obtaining permanent residence or citizenship in another EU country and then exercising your right of free movement within the EU.
I would be happy to detail if you pick one of these situations which seems most appropriate to you. Note that, in the case of the work option, you would have to have a secure job offer / employer in Denmark.
I hope my answer was useful and look forward to your further questions or your rating, which is essential to my activity.
Dr I L Vlad