score:3
First, in Europe (at least in France) you need specific visa to work, such as the Work & Holiday visa.
As for the possibility to travel and work in other Schengen countries while holding a Work & Holiday visa, it seems that it's not allowed, according to this page (in French).
To be precise, Schengen countries offer more freedom to other Schengen countries' citizens only. For other people like your friend, you have to individually check each country's policies. But from what I imagine, your friend can get a Work & Holiday visa in one country and visit the rest (if Singaporean only need a passport to visit a country).
A good idea would then be to pick a country either where salary is good or a central country from which it is easy to travel around.