For both US citizens and Danish citizens, no visa is needed in advance to enter the UAE. Immigrations officials simply stamp you passport when you walk through immigration in the airport and that grants you permission to stay in the country for 30 days.
To check if you need a tourist visa or not please refer to the following website.
Please also be aware that the UAE is very strict about the kinds of medications you can bring into the country.
Here is a list of the restricted medications for the UAE
I will not share exactly what for privacy reasons, but we were carrying a 4 month supply of a certain medication and were very heavily questioned about it even though the medication was not listed on the restricted list and we were traveling with the original labels, the prescription and a doctors note.
It all ended fine and we were allowed to enter the country with all of the medication but it was nerve wracking for a while.
If you will be traveling to the UAE with medication double check this to be sure your medication is not confiscated.