M you control everything-ISO, aperture and shutter speed. You have to center the meter ( usually ) to get a good exposure. This is where you have the most control of how you image will look.Depth of field, a little under or overexposed, blur or freezing of motion, lower noise with lower ISO.
A - you choose the aperture,( and ISO unless you have the camera set to adjust it) and camera chooses the shutter speed. Use when your main focus is on controlling your depth of field. As you change your aperture, the camera will adjust the shutter speed to make an accurate exposure.So I would think this is what you mean by shutter speed on auto??
S -you choose shutter speed,(and ISO unless you have the camera set to adjust it) and the camera chooses aperture. Use when you know you need a certain shutter speed, usually to stop action or help control camera shake( handholding)As you change your speed, your camera will adjust the other settings to make an accurate exposure.