All certified airplanes are designed to have a (very slight) spiral instability. The opposite case, dutch roll instability is not acceptable. This will make a plane that is trimmed out straight and level and then left alone without pilot, autopilot wingleveler or other automation to slowly enter a spiral dive with gradually increasing bank angle and sink rate.
Spiral instability is achieved by having the lateral tail area large enough so when a wing drops slightly by external disturbance and the plane starts to slip in the wing low direction, the tail will turn the plane in the same direction as the wing drop. This is a gentle and harmonic movement, it feels natural to the pilot. If uncorrected it leads to a spiral dive at increased bank angle and speed.
Dutch roll is the result of to small tail area, so when the plane slips for instance to the left, the nose will go right and up. The speed will go down, and if uncorrected will lead to a stall.
It is not the case that a plane left unattended (no pilot or autopilot on duty) will largely stay on course and altitude until the fuel runs out. If a plane was stable by itself, flight in low visibilty would not be a problem.
One famous example is the John F. Kennedy Jr. accident.