What is an easy way to see what routes are available from a given airport?

I am using Wikipedia for this. Each english wikipedia listing of an airport lists serving airlines and their destinations. So if you want to know where you can go from Hannover airport, just type Hannover (HAJ) airport wikipedia in google and you will be directed towards its listing in wikipedia, with a section on Airlines and destinations.

I haven't found an airport yet, that is not listed this way on Wikipedia.

Best site that I know of for this is:

Airline Route Maps

With over 1000 airline route maps, you could click on say, Oceania, and look at Air NZ's Global map. However, this is on a by-airline map really.

However, down the bottom is the search. Type in say 'Auckland' and it'll show you all the airlines that seem to have flights going through Auckland :D So it's a pretty good starting point.

Kinda annoyed really that I hadn't found it a couple of months back when I was trying to answer the question on which airlines fly from Mongolia...

I use a more pedestrian approach for that. You can usually find routemaps and/or timetables on the airport's website. Moreover you will find the names of the companies operating at the airport. Then I proceed from there.

Example: http://www.hannover-airport.de/

If there is no such site or if it does not work or if it is not available in a language I understand, I try Wikipedia as a backup. There are plenty of airports having an entry on Wikipedia and there you can get an idea of the routes and the companies.

Example: http://en.wikipedia.org/wiki/Hannover-Langenhagen_Airport

Use www.routehappy.com to search flight schedules.

For example here is a list of all places you can fly direct from London

(note: I work for Routehappy)

You can use wolfram alpha for this sort of stuff (although it works best with US airports):

For example "all flights from KJFK" a great thing about this search is you can see a map of planes estimated about your current location so you know where the airplane is headed.

For more international destinations, http://matrix.itasoftware.com/ is what I use (actually, their android app).

Kiwi.com, momondo.com, adioso.com, and others allow “anywhere” as a destination.

At one time, the app from Azuon.com was the best thing I had ever seen. But the Mac version broke more than a year ago, and they show no interest in fixing it. Maybe it still works on Windows.