We’re already coming across a lot of data, from very different sources. Let’s try to keep of track of all this.
The United Nations High Commissioner for Refugees has a nice data portal, which sums up all its missions. For this event, the Mediterranean portal is by far the most useful, documenting routes and sea arrivals - as well as keeping up to date the number of dead and missing persons.
The Commission’s statistical agency keeps records of a bunch of things, including of asylum applications and first instance decisions in the EU-28. Here’s their September 2015 quarterly report, as well as a ton more data, and the full dataset (complete with its silly webapp).
The World Bank has a dataset running from 1981 to 2014 called “Refugee population by country or territory of asylum.” The data can be grabbed in XLS, XML, and CSV format.