Before moving to Copenhagen, I read a lot of accounts from foreigners living here who complained that the Danes were rude. I argue that "rudeness" is highly relative. How rude the Danes seem depends a lot on how friendly your home town is.
My home town? Not especially friendly. Let me describe it a little bit...
I grew up in a suburb north of New York City. It's close enough that folks who work in NYC can buy homes and commute to work. It's got its pockets of less well-off areas, but overall the whole county has a good bit of money. And one thing that I've learned from having lived in both poor and wealthy cities is that folks with lots of money don't always feel the need to be friendly. Conversely, when I lived in Buffalo, NY, which is known for being almost as poor as Detroit, I found that almost everyone I met was open and friendly.
So now, let me compare waitstaff and retail workers in my hometown to the same here in Copenhagen. Back home, most of them are teenagers and 20-somethings. It's tough to make enough money making minimum wage or collecting tips to afford a place to live there, so a lot of them are still living with parents. Think back to yourself as an awkward teenager or disgruntled 20-something, and imagine trying to put a smile on your face and be friendly to a bunch of folks your parents' age who make more money than you. So, you'd tend to get service that was lackluster, to say the least. Here in Copenhagen, people live a fairly private life, even when in public. Interacting with strangers on a daily basis doesn't come easily to a lot of Danes. That being said, even the folks working in restaurants or in retail shops make a living wage here, so they tend to be less irritated about having to be at work.
Sure, the waiters don't work for tips here, which means they don't feel the need to put in the extra effort to be nice. But what that also means is that when they are nice, it's much more genuine. And let's be honest, the tip money doesn't help the kids back home working at the local diner to be especially friendly.
I can certainly see how coming from a town where everyone's fairly nice to everyone else can make the Danes seem cold to the point of rudeness, but I've found that folks here are usually as nice to us as we are to them. They don't take terribly kindly to being talked down to, as they have a culture that says that everyone should be on the same level, but my experience has been that when I'm smiling and kind to waitstaff and shop workers, they're pretty much always as smiling and kind right back.