Signing bonuses are fairly common. It's not a number that is well-arrived at by picking at random. The figure should be based on a calculated value exchange: ie,"I'm not fully vested at my current company, and will be giving up $5k worth of stock options if I leave. I'd like to have a $5k cash signing bonus as part of any new offer I'd take to offset that."

Well said. It's all about finding some "reason" to have a bonus -- something given up to take the position. Like someone said above, it doesn't have to be a great reason. Just a reason.

How common is this in Europe? For example, is relocating to another country something that might be a good reason for asking a signing bonus? How do companies see this?

Note I'm interested in experiences from Europe, I think that there's lots of hiring "cultural" differences between the continents...

I don't think they are exactly common, but they are not rare. I tend to see them at bigger companies.