Usage Note: Stomp and stamp are interchangeable in the sense “to trample” or “to tread on violently”: stompedstampedto death; stompingstampinghorses. Only stamp is used with out to mean “to eliminate”: stamp out a fire; stamp out poverty. Stamp is also standard in the sense “to strike the ground with the foot, as in anger or frustration,” [to bring the foot down quickly] as in He stamped his foot and began to cry. In an earlier survey the use of stomp in this example was rejected by a large majority of the Usage Panel.
The American Heritage® Dictionary of the English Language, Fourth Edition