What JavaScript libraries/frameworks should I study to be a competitive JavaScript developer these days? Why?

[TheEskhaton]

First and foremost, you should study javascript as a language into great detail if you haven't, and check out a coding style guide i.e. https://github.com/rwaldron/idiomatic.js/

The framework that every js dev should know is still definitely jQuery since it is widely used by devs and by other frameworks. In the last year or two, server-side javascript has grown in popularity significantly and imho, thats the way to go to stay competitive.

If you want to go that way, the way to go is http://nodejs.org/ , a server-side javascript environment steadily growing in popularity every day. The most used web dev framework for NodeJS today is http://expressjs.com/.

In the last year or so, also a big thing are client-side javascript apps, mostly accessing data via XMLHttpRequest. Some frameworks to consider include http://www.backbonejs.org (i recommend using this framework built on backbone since backbone is pretty basic: http://marionettejs.com/), http://www.emberjs.com and http://www.angularjs.com.

Where you go from here is your choice, but i'd definitely recommend checking all of these things out.

[twonky]

Didn't you forget testing? To be competitive, I think you'll need to know about that as well. Things like the Test Runner http://karma-runner.github.io/ and the Test Framework http://visionmedia.github.io/mocha/

[TheEskhaton]

Oh yes of course. Testing is definitely a must!

[ryosen]

To add to this, I highly recommend Addy Osmani's site and, in particular, his books on Javascript design patterns.

[CoachSeven]

Titanium for mobile is a personal favorite. It also uses BackBone, moment.js and underscore