Now I'll be the first to admit I'm not "bleeding-edge" with my choices, and there are likely newer and better alternatives to the libraries on this list out there. However, the libraries on this list do what they do very well AND have solid developer support AND a healthy community behind them, which can't always be said for the library du jour.
So, without further adieu ...
jQuery
jQuery is so popular that for many on Stack Overflow it has become synonymous with "Javascript" itself. Between 60%-70% (depending on whose counting) of the most popular sites use jQuery, and jQuery is even influencing discussions on the next version of javascript. Even Microsoft has embraced jQuery.Why all the love? Essentially it's because jQuery makes Javascript better. jQuery's event hookups, AJAX calls, DOM manipulation, etc. all offer a superior API to the native browser calls, plus jQuery handles any browser inconsistencies. It's fairly safe to say that using jQuery will change how you program in Javascript.
Underscore
I started using Underscore because Backbone required it, but I quickly discovered that the library was a great replacement for many of the Mochikit functions that I missed. While jQuery provides each and map, Underscore offers each, map, reduce, find, filter, reject, all, any, min, max, ... you get the idea. Plus they are can be chained together, kinda like jQuery (only with slightly worse syntax): _(someList).chain().map(function1).reduce(function2).min().value();Other really great stuff from Underscore includes bind/bindAll, keys/values, extend, and the various is* functions. Underscore even has a decent templater (although it's no Handlebars), making it the true "missing features of Javascript" library in my book.
Backbone
Back when I wrote single-page Javascript I used to think that MVC libraries were overkill ... and I still think that, for simpler sites. But if you're going to write a "thick client", ie. a completely Javascript-powered site, a MVC library is pretty much a requirement.Backbone provides classes for models, views, and collections. Models hold data and can trigger events when that data changes, collections hold models, and views interact with the DOM to both reflect and set the data in models. The separation between models and views is hugely helpful when writing large applications.
Now the C in MVC normally stands for controller, but Backbone doesn't provide a controller class ... and that's ok, because you don't need one (your controller is basically just normal JS; it doesn't need extra functionality). What Backbone does provide however is a router class, which helps your controller handle all of the (pure-JS) "page" transitions in your app. Using the router a user can go to "/#foo" and Backbone will know to run your "handleFooPageLoad" function.
Handlebars
This is another "not so important for the little guy" library. If you're not messing with HTML that much, just plain old HTML files and jQuery code will work for you. However, if you're finding that your Javascript is full of strings of HTML, and if your designer has to pester you every time he wants to add a class or change a word of verbiage, you need to start using templates.With Handlebar's templates you can specify a template like "<a href='{{url}}'>{{linkText}}</a>", and then use a JS object like "{url:'www.example.com', linkText:'This is an example link'}" to get the HTML you actually want, ie. "<a href='www.example.com'>This is an example link</a>". The templates can either live in script tags in your HTML files, or in their own separate files; either way they're much easier for someone else (like your company's designer) to work with.
Quick side note: Backbone's views are very agnostic about how they render themselves, which means you can generate your elements however you want ... including using Handlebars. There are even Backbone libraries for combining the two (although I personally have never used them).
XRegExp
This is the most specialized library of them all, but it's still worthwhile for anyone who needs to do anything serious with regular expressions in Javascript. It adds a number of "missing features" from the regular expression functionality in JS, including named groups! Why does that matter?Well let's say you have (as I once did) a large line of text that needed to be parsed in to some 20+ variables. Normally you would either have to write a bunch of different regular expressions, or else write one giant expression that's difficult to work with because everything has to be referenced by index. With named group regular expressions you can "name" each group in the expression, which makes it easy to access exactly the chunk of parsed data that you want.
Named groups are just one of the many great regular expression improvements in this library. If you're using regular expressions a lot in your JS, I highly recommend checking this library out.
Require
I've only started using Require somewhat recently, so while I really like it I can't endorse it quite as heartily as the other libraries. Require basically provides a new way to "load" your Javascript files. One minor benefit is a slight speed increase, as the loader can request multiple files at once. The real benefit though is that using the loader forces you to expose your dependencies.Let's say you have a class Foo, and you want to change some aspect of it, only you're afraid that doing so might break some other code that uses Foo. Or maybe someone else wrote Foo and you're just looking to understand how it relates to your other classes. Because Require forces you to spell out your dependencies you can easily find out exactly which classes use Foo, or which classes Foo itself depends on.
It's a little hard to explain why this is so great, since it sounds just like extra meta-data. Once you start using it though you appreciate how you can tell for certain where every variable in a file is coming from, or how you can recognize that a class has grown to depend on another class which it really has no business being involved with at all.
So that's it for my library breakdown. Hopefully I've introduced someone out there in teh intarweb to a new library, and maybe someday I'll even take a more in depth look at what these libraries have to offer. If I can get myself blogging more regularly that is :-)