How can I unit test client-side jQuery/JavaScript code inside ASP.NET MVC?

You could take a look at QUnit. I am not sure whether it's or it will be endorsed by Microsoft, but it's what's used to unit test the jQuery framework itself.

I use the following frameworks for JavaScript unit testing:

QUnit

As for me, it is the best choice if you want test your jQuery functions logic. It was developed by the jQuery developers and works with it well.

Jasmine

This is a BDD framework. It is very useful when you write your tests as specifications for your application.

Both frameworks can be integrated in the test explorer of the Visual Studio 2012 using Chutzpah Test Adapter for Visual Studio 2012.