It is actually very simple to eliminate singletons: just force the components to ask for what they need in the constructor or via setters or via inject*() methods, instead of looking up a singleton trough a static method only to obtain a reference. Once this fundamental decoupling is achieved, the hard part is tackling the construction problem.

In the first part of the two, he looks at a Zend Framework example of how to replace a certain piece of code with injected instances of required objects. In the second part he refines it down a bit more and only inject/create the objects and resources as needed.