Here is a list of good JavaScript libraries I prefer or find needing a lot.

Web Development

• Modernizr: Detects what features the visitor has in his browser.
• lodash underscore: Functional programming patterns (underscore version).
• Handlebars: String templating engine. Just as good as Mustache. I use whichever my other libraries depend on.
• Mousetrap: Keybindings to a website.
• d3: Creating data visualizations, charts, maps etc.
• Headroom: Show the page header only when scrolling up. Not sure about IE9 support.
• Nanobar: Lightweight progress bars.
• progressbar.js: Robust progress bars.
• scrollReveal: Show elements as you scroll down.
• ifvisible.js: Check if user is currently looking at the page.
• countUp.js: Shows running numbers.
• Moment.js: Manipulate and show times and dates.
• Howler: Play audio on websites.
• FatPixels: Draw pixel graphics.
• hammer.js: Multi-touch gestures.
• skrollr: Parallax scrolling and scroll animations.
• Tether: Absolutely positioned elements attach to elements in the page efficiently

Provide Structure

These libraries cannot be used together and usually define how you structure your code.

• React: Rendering library for web apps. (V of MVC)
• Swarm: Syncronized data objects for web apps. (M of MVC)
• Backbone: Provides tools for creating structure for web apps.
• Thorax: Framework on top of Backbone.
• AngularJS: Provides structure for web apps.

jQuery:

• jQuery: A bloated but powerful JavaScript for DOM manipulation etc.
• FitText: Resizes your headlines for all resolutions.
• FitVids: Responsive video width.
• BigVideo: Full screen background videos.
• jRumble: Shake HTML elements.
• Midnight: Change a fixed header as you scroll down.
• Slick: Carousel plugin.
• OriDomi: Fold up HTML elements like paper.
• FocusPoint: Reposition and resize big images based on a focus point. Great for full screen backgrounds.
• Shuffle Images: Show multiple images by moving curser around.
• Particleground: Background particle system.
• HalfStyle: Apply a style only for a part of a line or character.
• One Page Scroll: Create websites that scroll down one view at a time.
• multiScroll.js: Divided multi-scrolling websites.

Node

• async: Async utilities.
• Nightmare: High level wrapper for PhantomJS to automate browser tasks.