jQuery
jQuery is a fast, small, and feature-rich JavaScript library. It makes things like HTML document traversal and manipulation, event handling, animation, and Ajax much simpler with an easy-to-use API that works across a multitude of browsers.
YUI
YUI is a free, open source JavaScript and CSS library for building richly interactive web applications.
Zepto
Zepto is a minimalist JavaScript library for modern browsers with a largely jQuery-compatible API. If you use jQuery, you already know how to use Zepto.
Dojo
A javascript toolkit.
MooTools
MooTools is a compact, modular, Object-Oriented JavaScript framework designed for the intermediate to advanced JavaScript developer. It allows you to write powerful, flexible, and cross-browser code with its elegant, well documented, and coherent API.
Minified.js
Minified.js is a client-side JavaScript library that's both powerful and small. It offers jQuery-like features (DOM manipulation, animation, events, HTTP requests) and utility functions (collections, date&number formatting, date arithmetic, templates) with a single, consistent API.
Prototype
Prototype takes the complexity out of client-side web programming. Built to solve real-world problems, it adds useful extensions to the browser scripting environment and provides elegant APIs around the clumsy interfaces of Ajax and the Document Object Model.
Snap.svg
MV*
AngularJS
HTML is great for declaring static documents, but it falters when we try to use it for declaring dynamic views in web-applications. AngularJS lets you extend HTML vocabulary for your application.
Backbone.js
Backbone.js gives structure to web applications by providing models with key-value binding and custom events, collections with a rich API of enumerable functions, views with declarative event handling, and connects it all to your existing API over a RESTful JSON interface.
Vue.js
Vue.js is a library for building interactive web interfaces. It provides data-driven components with a simple and flexible API.
Ext JS
Ember.js
A framework for creating ambitious web applications.
Knockout
Simplify dynamic JavaScript UIs with the Model-View-View-Model (MVVM) pattern.
Durandal
MVC? MVP? MVVM? Yes. Messaging, navigation, modals? Check. Durandal has the features you need to build whatever apps you can imagine.
React
A javascript library for building user interfaces.
Ractive.js
Ractive.js is a template-driven UI library, but unlike other tools that generate inert HTML, it transforms your templates into blueprints for apps that are interactive by default.
Mithril
Mithril is a client-side MVC framework - a tool to organize code in a way that is easy to think about and to maintain.
Utilities
Underscore.js
Underscore is a JavaScript library that provides a whole mess of useful functional programming helpers without extending any built-in objects. It’s the answer to the question: “If I sit down in front of a blank HTML page, and want to start being productive immediately, what do I need?” … and the tie to go along with jQuery's tux and Backbone's suspenders.
Lo-Dash
A utility library delivering consistency, customization, performance, & extras.
Moutnew
MOUT is a collection of modular JavaScript utilities that can be used in the browser as AMD modules or on node.js. It provides many helper methods similar to those found on other languages standard libraries (ie. Python, Ruby, PHP).