Getting the CPR number was easier than I thought and took just three hours; however as I say I was very lucky. If you are considering moving here with no significant ties to Denmark such as Danish family/fiancé/etc., then I suspect it will be more difficult. Let me go through what I did and what the more difficult parts were, step-by-step.
Your first port of call should be Borgerservice (here), which I guess translates as Citizen Service. The international Services desk however is only open on Wednesdays and Thursdays at certain times - see this link for opening times and what you'll need to bring. My one tip is to get there as close to opening time as you can with all the documents you can possibly arm yourself with - when I got there an hour after the international desk opened there was already a fairly lengthy queue.
Documents I took with me that were needed:
- A complete registration permit form - this is needed before you get the CPR number, even if you are EU - the application form is here
- My fiancé with his bank statements - as I am now unemployed I came here under 'sufficient funds', which means I have to prove that I or someone can fund me during my stay in Denmark, which my fiancé can. I imagine that this would be absolutely vital if you are self-employed.
- Two passport pictures - they provide this at Borgerservice but you will need these before you queue up at International Citizens Service so perhaps get there before the International Citizen desk opens in order to get these done (fee applies)
- Handwritten and signed letter from the person responsible for loaning me the flat (happens to be my fiancé in this case)
- Full driving licence with paper counterpart
- Proof of residency in Denmark such as rental agreement
- Letter from employer
I must reiterate again that I was very lucky in that I have a Danish fiancé who is familiar with the Danish systems and is able to vouch for me; therefore before going to the Borgerservice office I would most definitely recommend giving them a call to see exactly what documents you need before you visit them in person. There's little point filling out the application forms before you leave for Denmark as this will not save any time however it would definitely be worth collating all documentation such as the above (plus anything else you feel would be useful) prior to departure, to save hunting for it when you get here.