At Clarity, we chose to do things a bit differently knowing that we wanted to have an optimal experience for mobile devices. I would say that there's no one framework that specifically allows us to do that and that it's more about the mindset than the frameworks.
With that said, your question was specifically for frameworks, so here's a few. Even though we don't necessarily use all of these at Clarity, we probably should given some of the things that we've learned over time:
* http://getbootstrap.com/ - Great CSS library to build responsive websites.
* To handle the complexity of the client side application, a good MVC framework is in order
** We use http://spinejs.com/ here at Clarity.
** I've also heard a lot of good things about http://emberjs.com/ and http://angularjs.org/
** I find that for smaller projects, http://backbonejs.org/ is a good choice as well.
* One thing that you'll notice on mobile browsers is the delay in click events being triggered. Google has a great article describing the problem: https://developers.google.com/mobile/articles/fast_buttons, and https://github.com/ftlabs/fastclick seems to be a popular implementation to solve this problem.
A short list of tools to work across browsers, devices, and platforms.
* Bootstrap (http://getbootstrap.com): A front-end framework whose UI components work across all browsers and devices. They have an extensive plug-in library.
* Foundation (http://foundation.zurb.com): Bootstrap's main competitor. They have a different grid system for placing components.
* jQuery (https://jquery.com): The defining library for interacting with the DOM across browsers and devices.
* Modernizr (https://modernizr.com): A tool to detect browser capabilities rather than check for a specific browser.
Pure JS Frameworks:
* React (https://facebook.github.io/react): Technically a library. Facebook built React to simplify UI building and provide a virtualization layer over the DOM.
* Backbone (http://backbonejs.org): An MVC framework. It's barebones and malleable.
* Angular (https://angularjs.org): An MVC framework that integrates through directives embedded in your HTML.
* Ember (http://emberjs.com): Yet another MVC framework.