Come on, guys! Blender doesn’t bite… 
It took me 5 minutes to build a simple armature and to experiment the various “spaces”.
I added a Limit Rotation constraint to the Bone.
World Space: As the name suggest, limits, movements and rotations use the global coordinates.
Local Space: Limits, movements and rotations use the local coordinates of the Bone.
Local with Parent: The same thing than Local but using the local coordinates of the Parent.
Pose Space: This time, we use the local coordinates of the Root of the Armature.
There can be some overlapping. If the parent is also the root, “Local with Parent” and “Pose Space” are the same… And they are the same than “World Space” if this parent/root bone isn’t rotated.
The use for “Local Space” is obvious in a skeleton. “World Space” is useful to make things hang or float while respecting the gravity. I might find a use for “Pose Space”… but “Local with Parent” puzzles me. 
Now, don’t ask me what happens when you convert coordinates from one space to another, i.e. when the modifier has 2 dropdown menus for the coordinates. :spin:
Any way, all I really know for sure is that I’d like to have an option for all the modifiers to be in “Local Space” by default since that’s the one I use in 99.9% of the cases. :eyebrowlift: