As someone who's opposed to conscription, I have been doing some research on the topic in various countries.
What I find interesting is the issue of defining what actually counts as conscription. Many countries clearly either do or don't have it, but there's also plenty of countries where it's not so clear.
Consider Denmark. Denmark's constitution makes it clear that all male adults who are physically able are required to complete military service:
Every male person able to carry arms shall be liable with his person to contribute to the defence of his country under such rules as are laid out by Statute Constitution of Denmark
Reading this, you could be forgiven for assuming then that the issue is pretty clear cut. Denmark clearly has conscription and everyone is required to participate.
Day of Defence
On their 18th birthday, every male is drafted to the 'Day of Defence', where they are given an introduction to the Danish military. And his lies the first hurdle - if you aren't physically fit and don't mind testifying to that effect, you can get out of attending.
Assuming you believe yourself to be physically fit however, you attend the Day of Defence expecting to be drafted into the military. But of course, as part of the Day of Defence you are subjected to health tests, and in practice they'll declare you unfit for pretty much any reason they can think of.
Picking numbers from a hat
But let's say everything is just fine and they declare you fit or partially fit for military service. What happens then? Well they make you draw a number between 1 and 36,000. If you draw a number below 8,000 you should, in theory, be drafted into the military.
Of course, there's another hurdle still. Remember how some people are simply declared 'partially fit'? They're immediately disqualified, regardless of which number they pick. Of course, that leaves thousands of people still drafted into the Danish military - surely then, they have conscription?. Now, you certainly wouldn't be wrong to assume this, but there's another important detail.
Denmark has an active volunteer force. Of the men selected, 99.1% were volunteering as of 2014. What does this mean? Well, the military only needs so many people, and apparently they have enough volunteers anyway. In fact, in 2014 only 19 men were actually compelled to join the armed forces.
What is interesting though is that a recent survey found that around 2/3rds of Danes do still oppose conscription, though it seems unlikely that the government would be in any huge rush to end it. Why bother? They've got plenty of volunteers and there's still an argument to be made that men could be conscripted should the volunteer force prove insufficient in the future.
So does Denmark have conscription?
Well yes. It does. But then, so does the United States, at least in a legal sense. In the United States, all male citizens are required to register for the selective service. This hasn't been used since the Vietnam war, but the mechanism is still there. Now, obviously Denmark's system goes beyond that. They have their 'day of defence' and they do conscript people into the armed forces. But at the same time, it's so comically easy to get out of conscription and so many people seem to get out of it anyway (again, only 19 people were conscripted in 2014) that it's easy to see how Denmark's system is certainly very different to that of say, Israel.
Denmark is only one example but there's plenty of other countries where the question "do they have conscription" is perhaps not quite as easily answered as one might hope.
A European Army?
Regardless of your views on an EU-wide army, it does appear to be a potential solution to the issue of conscription within the EU. Looking at a map, you'll notice that the majority of EU countries that still have conscription are, perhaps unsurprisingly, countries with smaller and sparser populations, and thus less volunteers into the armed forces to begin with.
Perhaps if there was an EU-wide military, this would become un-necessary as soldiers pooled from throughout the European Union could make up the shortfall in countries with insufficient volunteers.
The only problem is marketing this to countries where national pride outweighs the potential benefits of an EU-wide army.