The first fear that comes to mind is snakes....I've been irrationally afraid of them since the age of five. I've managed to partially overcome it by being around non-poisonous varieties...I'm now even comfortable holding those for a short time.
Another fear is rejection, whether in my personal life or my writing life. The best way to overcome that one is to just put myself, or my writing, out in the world, and take a chance. I'll be doing just that next month when my revised book of poetry is released.