Should private helper methods be static if they can be static

It might result in slightly smaller bytecode, since the static methods won't get access to this. I don't think it makes any difference in speed (and if it did, it would probably be too small to make a difference overall).

I would make them static, since I generally do so if at all possible. But that's just me.

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EDIT: This answer keeps getting downvoted, possibly because of the unsubstantiated assertion about bytecode size. So I will actually run a test.

class TestBytecodeSize {
    private void doSomething(int arg) { }
    private static void doSomethingStatic(int arg) { }
    public static void main(String[] args) {
        // do it twice both ways
        doSomethingStatic(0);
        doSomethingStatic(0);
        TestBytecodeSize t = new TestBytecodeSize();
        t.doSomething(0);
        t.doSomething(0);
    }
}

Bytecode (retrieved with javap -c -private TestBytecodeSize):

Compiled from "TestBytecodeSize.java"
class TestBytecodeSize extends java.lang.Object{
TestBytecodeSize();
  Code:
   0:   aload_0
   1:   invokespecial   #1; //Method java/lang/Object."<init>":()V
   4:   return

private void doSomething(int);
  Code:
   0:   return

private static void doSomethingStatic(int);
  Code:
   0:   return

public static void main(java.lang.String[]);
  Code:
   0:   iconst_0
   1:   invokestatic    #2; //Method doSomethingStatic:(I)V
   4:   iconst_0
   5:   invokestatic    #2; //Method doSomethingStatic:(I)V
   8:   new     #3; //class TestBytecodeSize
   11:  dup
   12:  invokespecial   #4; //Method "<init>":()V
   15:  astore_1
   16:  aload_1
   17:  iconst_0
   18:  invokespecial   #5; //Method doSomething:(I)V
   21:  aload_1
   22:  iconst_0
   23:  invokespecial   #5; //Method doSomething:(I)V
   26:  return

}

Invoking the static method takes two bytecodes (byteops?): iconst_0 (for the argument) and invokestatic.
Invoking the non-static method takes three: aload_1 (for the TestBytecodeSize object, I suppose), iconst_0 (for the argument), and invokespecial. (Note that if these hadn't been private methods, it would have been invokevirtual instead of invokespecial; see JLS §7.7 Invoking Methods.)

Now, as I said, I don't expect there to be any great difference in performance between these two, other than the fact that invokestatic requires one fewer bytecode. invokestatic and invokespecial should both be slightly faster than invokevirtual, since they both use static binding instead of dynamic, but I have no idea if either is faster than the other. I can't find any good references either. The closest I can find is this 1997 JavaWorld article, which basically restates what I just said:

The fastest instructions will most likely be invokespecial and invokestatic, because methods invoked by these instructions are statically bound. When the JVM resolves the symbolic reference for these instructions and replaces it with a direct reference, that direct reference probably will include a pointer to the actual bytecodes.

But many things have changed since 1997.

So in conclusion... I guess I'm still sticking with what I said before. Speed shouldn't be the reason to choose one over the other, since it would be a micro-optimization at best.

My personal preference would be to declare them static, as it's a clear flag that they're stateless.

I prefer such helper methods to be private static; which will make it clear to the reader that they will not modify the state of the object. My IDE will also show calls to static methods in italics, so I will know the method is static without looking the signature.