All your usual high street type shops and hypermarkets abound, but many of the independent shops are classier and tidier with more helpful, friendly staff than British equivalents. One impressive feature of Danish shops is that they offer to wrap purchases for you for free if it's a present. This is especially handy come Christmas time.
Some things are similar prices to
other countries, even with 25% sales tax, but when it
comes to books, cds and other items you might get a bit shocked. Thanks to the internet, however, you can shop abroad and buy all those items you want at more familiar prices using Ingen Moms.
A popular destination for Danes is to pop across the border to Germany to stock up on beer, wine, spirits, soft drinks and certain food items. You'll see this by the large number of stores positioned just on the German side of the border, whereas the Danish side is full of sex shops for Germans to stock up on their, ahem, key requirements. When we did this we also found electricals and light bulbs to be among the other items that are cheaper. Not necessarily a great bargain overall after you've figured out the cost of driving down, but definitely worth a trip out for the day.
Finally, a fairly irrelevant tip. In my home town all the supermarkets are very poor at clearing out of date food off the shelves. My tip for a Monday morning is to pop into Føtex, look for something that's out-of-date - the dairy section is always a good bet, buy it then take it to the customer service desk and complain. You should get either a bottle of wine or a box of chocolates.