Thousands of times I hear: "WHEN USING A TRIPOD SET THE STABILIZER ALWAYS OFF". An image is worth 1000 words:
Shots done with an E-P5, 135mm, 1/1,6 s @ f/5,6, ISO200, rear wind over the tripod+camera combo was simulated with a fan.
Intuitively I always thought the IS could improve shots on a tripod when the conditions are not stable (e.g. wind or ground vibrations). When shooting on a stable environment (indoors, outdoors in absecene of wind), there is no practical difference between using the IS or not, so you just don't need to care. But when the conditions are the opposite the IS helps to improve image sharpness.
So the "WHEN USING A TRIPOD SET THE STABILIZER ALWAYS OFF" myth is false, I'd rather say "WHEN USING A TRIPOD SET THE STABILIZER ALWAYS ON", and by extension in practice: "LEAVE THE STABILIZER ALWAYS ON".
Regards