Though I'm fortunately not a person who has to ride the trains on a daily basis in and around Copenhagen, I am still fairly confident when I say that in Denmark, DSB, the Danish train operator, is one of the better train operators out there. The trains generally run on time, their schedules are all coordinated so making connections is generally very simple, and they seriously can get you pretty much everywhere you need to be in Denmark (at least when coupled with a bus connection at the end of the line in the case of the more remote places you're trying to reach).
Case in point: I've never lived in a country, or even heard of a country, where you can take public transportation to reach an off-the-beaten path summerhouse. But I've been able to do it to get to more than one summerhouse in Denmark.
Anyway, the latest addition, is the ability to see where your train is on a GPS maps displayed as part of the timetable displays at many stations, which I suppose could be nice to know if it's running late. Not that it will help you get to where you want to be any faster, if they are late, of course.