The most common misconception.

I've had at least 50 people tell me the following:
(Including more than one scientist with a hard science PHD)
Why not put your rocket engine on top so the vehicle will be naturally stable.

This is wrong. It jives with our natural experience of the world, but it is wrong.

A rocket has no natural preference for thrusting UP. The rocket will thrust in whatever direction you point it. This is unusual in out natural experience. It you hang something on a rope the rope has the natural tendency to dangle straight down and only apply force in the up direction. If you attach something to a balloon the balloon only pulls straight up.

So given that the rocket will thrust in whatever direction its pointed and has no natural tendency to point up the rocket can't add any stability.

So if the motor can't add stability is there anything you can do for gravity to add stability? If the gravity field was uniform it would operate on the center of mass appling no torque to the vehicle and thus no stability. On the surface of the earth this is basically the case. The gravity field varies with the square of the radius. In orbit this slight difference can be used to stabilize a spacecraft, but on the surface the effect is so slight as to be undetectable. (On a 2meter high object gravity is 0.999999969 as strong on the top as it is on the bottom)

Our intuition is not completely wrong with respect to where the rocket motor goes, if it's center of thrust is not pointing through the center of mass then the rocket engine will apply a torque and the vehicle will spin. The current unreasonable rocket design has four engines that must be balanced to keep the vehicle from spinning, so in this sense a single rocket motor pointing toward the center of mass would keep the rocket pointed in a constant direction, but that direction would not be preferentially up without some kind of external control.