Q:

What methods are available in the standard controller?

Motivation
I'd like to use a lot of functionality from the standard controller and only add some additional fields.
For example, Lead standard controller apparently provides filtered Converted Statuses, returning only those that are configured as allowable. 
Question
How can I inspect what methods are available on the standard controller?

A:

The methods on the StandardController are defined here, this is used when dealing with a page showing a single record. When dealing with a list of records you use the StandardSetController, documented here. 
You can setup the StandardSetController to expose the List View information and results via the setFilterID and getRecords methods (you might find this question useful to see these methods in action). Also a good example of using it can be found here "Using List Views with Standard List Controllers".
Lead Filters: I am not sure which list view filter your referencing, it don't see an obvious one in my DE org. However you can of course create your own through the standard UI and using the approaches referenced above access the records your filter returns.

