Somehow I cant apply any constraint to any object in my current project file. Adding new objects or new scenes doesnt change anything. Creating a new file is no problem, then the menu is back there again.
So, something with my file must be broken. Any ideas?
Having you tried resetting your preferences?
Well good idea! I tried it, but it doesnt work - which makes sense as the other files dont have that problem 
EDIT Ok, that seemed to be some GUI bug - I just “deleted” the properties panel and added it again and voilá: there we have our constraints again. Really strange, never have seen a bug like this in Blender yet 