What are the available $A JavaScript functions in Lightning?

Answering my own question. The Lightning Components Developer's Guide has the following:

The Aura object is the top-level object in the JavaScript framework code. For all the methods available in the Aura class, see the JavaScript API at https:/[mySalesforceInstance].lightning.force.com/auradocs/reference.app, where [mySalesforceInstance] is the name of the instance hosting your org; for example, na1.

So you can go to https://na15.lightning.force.com/auradocs/reference.app for example and see something like the following:

Here is some $A.util usage

// Find a component where you wan to apply the util
var myAttr = component.find("myAttr"); 
$A.util.addClass(myAttr, "add-class");

Similarly

getBooleanValue
hasClass
isArray
isEmpty
isObject
isUndefined
isUndefinedOrNull
removeClass
toggleClass