Hi Ovidiu, as your time permits I would recommend you learn as much as you can, even if you have no immediate plans to use these different frameworks. The reasons why you should do this are many but the top of the list would include: 1) continuing education is always a good idea; 2) great engineers tend to be found on these big projects and so learning from them can only improve your own coding; 3) you may be called on to use another project in the future and knowing the "ground floor" will greatly decrease your ramp-up time; and, 4) coding is much like writing in that as a general rule any good author will read more than s/he writes.
As for what's going on out there right now: Backbone is still being used on new projects and many older projects. (See Trello) It's not that difficult of a framework to grasp so I would start there if you've had no experiences with it. Learning the framework will also get you used to more advanced frontend methodologies.
Learn React. It's being used everywhere now, especially in enterprise level projects. It's super easy to grasp (you can learn the basics and begin coding within a day or two). Most of your time here will be spent on learning the hundreds of addons, mixins, and state-handling packages (like Redux).
Vue is a new framework and so not used widely (yet). It may or may not be picked up but it would be worth reading through the docs and learning the basics in case it does.
Angular 1 is dead so don't waste your time. Angular 2 is a monster (with Typescript) and will take the longest to pick up. It's also a mixed bag. A lot of companies are dumping it for React for this reason. Some others are picking it up because it does well at controlling logic across really large teams.
You can't go wrong with learning more even if you may not use a framework. Learning the underlying ifs/wheres/whys will teach you how to code better. :)
Update: I don't mean you shouldn't learn Angular 1 because, like many Backbond projects, there are a lot of Angular 1 projects our there as well. I mean I wouldn't seriously learn it to plan to develop with it.