The term "Mad Dog" was not a common one for any of these three famous Generals, however if you had to pick one, it would be William T. Sherman. As a brand new General early in the Civil War, Sherman was very stressed and not actually a very effective leader. He constantly overestimated the size of his enemies, and became sullen and whiny and timid in his operations. This was attributed to him being "insane" or having a "nervous breakdown" or "mad" at various times. He actually resigned his generalship for awhile, but then came across Ullysses S. Grant and the two emotionally walking wounded leaders with similar backgrounds took a deep liking to each other. They went on to win many famous battles together and eventually the war. Southerners also called Sherman Mad Dog because of his ruthless march through Georgia. At the end of the war, Sherman said of Grant: "I stuck by Grant when he was a drunk, and Grant stuck by me when I was crazy."
The term Mad Dog has been more appropriately used in modern times to describe the bravado associated with General James Mattis of the US Marine Corps, and later, as a Secretary of Defense under President Trump.