In Copenhagen I was strongly adviced not to leave my bicycle unlocked outside, for quick visit at a grocery store. Everyone I know takes their bikes home overnight.

Living in Copenhagen, I do not recognize that. I live in a city apartment on Vesterbro. We have maybe a hundred bikes parked in the backyard, nobody I know brings his bike home for the night, unless it is a really fancy racing bike. But yes, I do lock my bike when ever I really away from it.