Phoenix
This is the rarest core type. Phoenix feathers are capable of the greatest range of magic, though they may take longer than either unicorn or dragon cores to reveal this. They show the most initiative, sometimes acting of their own accord, a quality that many witches and wizards dislike.
Phoenix feather wands are always the pickiest when it comes to potential owners, for the creature from which they are taken is one of the most independent and detached in the world. These wands are the hardest to tame and to personalise, and their allegiance is usually hard won.
Elm
The unfounded belief that only pure-bloods can produce magic from elm wands was undoubtedly started by some elm wand owner seeking to prove his own blood credentials, for I have known perfect matches of elm wands who are Muggle-borns. The truth is that elm wands prefer owners with presence, magical dexterity and a certain native dignity. Of all wand woods, elm, in my experience, produces the fewest accidents, the least foolish errors, and the most elegant charms and spells; these are sophisticated wands, capable of highly advanced magic in the right hands (which, again, makes it highly desirable to those who espouse the pure-blood philosophy).