One of the legal modifiers you can use with method local inner classes is abstract.
For example:
public class Outer {
public void method(){
a
IMHO, this feature has NO real use. There's a couple of possible abuses, but there are many other ways to write bad code, you needn't learn this one. :D
Whenever you try to make use of an abstract method-local class, you need to define at least two concrete method-inner classes. This means you end up with a method containing at least three classes, the method gets quite long and that's quite a bad style.
You have to know this for the SCJP exam.
I really hope not. Method-local inner classes are already useless enough to be considered a corner case (you should understand them but probably never use them).
IMHO, a person asking this in an exam misunderstood Java badly. There can't be accessibility modifiers on a local class since (lacking method literals) the class can't be accessed from the outside anyway. There can be abstract and final modifiers, since there's no reason to forbid them. There are good reasons to allow them: orthogonality and the Principle of least astonishment.