Lækkerbar in Vesterbro is one of the better small natural wine bars in Copenhagen — a low-lit, unpretentious place with a short menu of charcuterie, cheese, and seasonal small plates to go with the wine. The list is mostly European natural producers, changed regularly, and the staff know the wines well enough to guide you without being tedious about it.
The neighbourhood around Vesterbrogade is worth exploring before or after: several bakeries, a good bookshop, and the kind of lived-in street that has not been fully renovated into something Instagram-ready. Lækkerbar itself is small — a dozen seats at the bar and a few tables — so arrive early or expect to wait.