Something you might not know if you've never tried to look but it is incredibly difficult to find an apartment in Copenhagen. From all the people I've spoken to, it takes between 3 and 6 months to find somewhere suitable to live. It's not that there aren't any apartments, it's that demand is incredibly high.
Probably the most popular website for apartment-hunting is BoligPortal. Like most similar sites, searching is free but in order to be able to contact the letting agent, you pay a subscription fee. Importantly, it is free to create saved searches and receive notifications of new apartments. Unfortunately, these searches only seem to run every 12 hours or so. It is quite common to receive a notification of an apartment that came on the market earlier that day only to find it was taken several hours ago.
On the plus side, BoligPortal have an unauthenticated (although not necessarily offical) API available.
Given all the above facts, it was inevitable that I'd end up making a program to solve the problem.
PushBolig - Pushbullet notifications for BoligPortal
I'm sure that there are already a dozen other applications out there that do the exact same thing but do any of them have such an appropriate name? It searches Bolig Portal and creates PushBullet notifications.
It's a simple, configurable Node.JS application that will search the BoligPortal API for the criteria you specify. Provide it with your Pushbullet Access Token and every time it finds a new apartment that matches, it will send the link to your devices. There's also a local database to make sure it doesn't send the same link twice.