I know Iceland still does! In Denmark it seems to have died out in the mid 1800's. Looking through the church records, the first thing I noticed was that the daughters were now "...sen" instead of "...datter", and their children took the fathers surname. There is also the confussion of a surfix, usually denoting a place or occupation, sometimes this would go down the generations and sometimes not!
I could also see that some familes obtained permission (usually from the bishop or even the king), to use the surfix as their official surname - even today the surnames that end in ...sen are thourght to be common, and people with rare surnames are very possesive of them. By law if there is under 2000 people who have a particular surname then it is protected, and not everyone would be able to use it as a surname. (So my surname's safe then

The latest change in the law has made it possible to use Patronym's again, but I've not heard of anybody doing so.