When you arrive to Denmark and before making any contract with any company just buy SIM card from Lebara from 10 danish kroner it is about 2 us$ that gives you a phone number and a credit for 10 us$, you can buy them from kiosks, 7/11, shop places like Netto, Fakta etc or super markets. You can use it and recharge it, it is not binding, no contract no headache, after while when you are in the country you can research for monthly contract offers and find the reliable solution. Most of monthly contract require Danish ID number (CPR number) and they are binding for minimum of 6 months.
Price gets down every year OR the same price and you get more content in it.
Example CBB offer mobile internet connection from 49,- danish kroner pr month with 1Gb or 89,- danish kroner for 10 hours talk, 10Gb 4G mobile internet.
Since I posted this answer a lot of things happened, because of hard competitions between telecommunication companies in little Denmark. The company behind CBB and Bibob called Telenor, now has merged both companies and has discounted the old offer of 49,- I mentioned earlier.
So I do not want to make Ad for this or that company but I will write my personal experience and it is always good to be aware of what happening by asking friends, check the internet or ask in forums/community sites like Travel.
Here is a link to comparison table of different telecommunication companies in Denmark the prices might not be up to date, but will give you some estimates and idea where to search.
Regarding CBB, cheapest offer right now is 69,- that includes 2 hours talk and 1Gb internet connection, but I personally have switch to Oister they have better internet coverage and bandwidth in my location even thus CBB has 4G connection it fill like they have had bad bandwidth in my area. Oister prices starts from 69,- month.
So some time it is not only price but the quality you get and depending on your location. Some companies has better coverage over all in Denmark, some has better in big cities etc. some could have bad coverage in specific locations etc.
My best advice start with non binding SIM cards, and when you fill your self confident, than go a head a try one of the offers, and Remember to ask if they have English contract, otherwise ask some body you know to help you reading the roles and disclaimers before signing in. BTW. most companies offer online ordering of new number, but as I said that required you identification (CPR number) in Denmark.
Finally any one what to know or ask more, you are welcome to comment and I will be glad to come with my knowledge.
New update: I have today checked Oister and they have 49,- Kr. offer. that includes 1 hours talking and 1Gb internet check
Oister charge 10 kr. pr account invoice, if you and have 2 phones number under same account then it is still 10 kr.