Answer: It would destroy the Earth's surface.
I remember reading this questions in What If? by Randall Munroe. Great book, I suggest you check it out. Anyway, the answer. The Earth is revolving as well as spinning on its axis at the same time. This basically means that thee atmosphere is also spinning at the same speed. But due to the frame of reference, we don't notice anything. If the Earth suddenly stops spinning, then the atmosphere, going according to the first law of motion will still be spinning at the same speed. This would produce supersonic winds at such a scale that it will be compared to an atomic explosion. Anything not in a nuclear bunker will probably be ripped apart by the force of the wind.