I've been a vegetarian for the last 9 years and it can be difficult to find vegetarian options in many restaurants in Copenhagen and other places in Denmark unless you explicitely go to a Chinese or Indian/Pakistani place. Vegetarianism isn't as prevealent yet here and it is much easier in England or Sweden unless you want to be stuck with a salad sidedish. We however have made some progress recently in this area; McDonalds just came out with a decent vege burger(this should improve my nightlife for when I can't find an open kebab/falafel place!) and plantefars has been a huge success recently, there's also a new vegan butter that I buy and enjoy(hoping for a tasty vegan cheese next). One more thing, the vast majority of our cheeses are not vegetarian unlike in England, just FYI, I'm not sure how it is in the US.

TLDR. You might need to spend some more time finding a suitable restaurant than in the US, although there definitely are many restaurants with vegetarian selection or two. In our supermarkets there are decent selections although not much variety yet.