Wand core: Basilisk Horn (Canon)
As basilisks are so rare, and actually slaying one even more rare, basilisk wands are very, very rare. It is unlikely to ever find new wands in a shop, and most that currently exist are heirlooms passed down in prominent pureblood slytherin families. When a new basilisk horn wand is made, it will certainly choose a parselmouth.
Unbelievably powerful and incredibly forceful, basilisk horn wands excel in all forms of magic, especially those magics that petrify, jinx, hex and curse. These wands are strongly bent towards dark magic and truly lethal with it.
The ideal owner of a basilisk wand is never dim or a shrinking violet. They are ambitious, cunning, brilliant and often exude charisma and power. These wixes make incredible leaders and planners. Entirely self sufficient and pragmatic, they know what they want and they are ruthless in their determination to get it. They often want to be remembered for their great deeds and prodigious skills, and so constantly work to be the best at everything they do.
Wielders of Basilisk horn wands often have a natural resistance to poisons.
It is best that a parselmouth be the owner of a basilisk wand, as this core is highly intelligent and may try to take charge in spellcasting, or even fire off curses without the owner's consent. Those gifted with parseltongue have a natural rapport with basilisk wands and are able to better control them.
"I really, really, really want to make a wand with this core. The school refuses to let me dig up the one buried in the grounds." - January Hawkins, resident wandmaker, Ilvermorny school of witchcraft and wizardry
(Image from Harry Potter wiki)