Just as modern darts do, medieval military darts have fletching of some kind. A dart just looks like a short arrow with a thicker shaft. (In fact, the arrow is thought to have developed from the dart.)
Being fletched missiles, they fly straight without spinningtumbling, just like an arrow does, though perhaps with a bit of fishtailing immediately after the throw as the fletching cancels any sideways momentum caused by the arm's motion. They don't spin end-over-end unless it broke mid-throw.