In a certain photograph you can see the flag waving. That's impossible in a vacuum.

It's always amusing to hear assertions of motion based on the evidence in a still picture.

It would seem that this question needs no rebuttal. But we should clarify that the apparent waving "motion" in the still photos is the wrinkles remaining from its packing. In earth gravity the weight of the fabric itself is often enough let wrinkles "hang" out. But because the flag was made from a very light nylon [Platoff93, note 10] which is even lighter in lunar gravity, the force of the wrinkles wanting to stay wrinkled overcomes the force of gravity for longer.

Here (Fig. 1) is an example of a flag that appears to wave. It's worth belaboring a trivial point to emphasize that observers will tend to "fill in the gaps" in their perception by applying past experience. The still photo doesn't actually show motion -- no still photo can. But the visible ripples cue our recall of all the other flags we've seen where rippling is caused by wind. And if we are not conscious of this extrapolation, we may strongly convince ourselves that we have indeed "seen" the detail provided by our memories.

This is why great care must be taken in interpreting Apollo photos. We cannot allow our prejudices of the behavior of objects in air and strong gravity to influence our interpretation of behavior on the moon.

In the video coverage you can see the flag waving. That's impossible in a vacuum.

The simple answer is inertia. The Apollo flag assembly starts with a telescoping tubular pole shoved vertically into the lunar soil. But the resemblance to terrestrial flag arrangements stops there. On earth we attach flags to the pole at the top and bottom corners. And the same would work on the moon, except that the flag would hang limply without ever being visible for what it was.

Fig. 1 - The flag assembly prior to packing. Note the peculiar way the flag was folded. This produced a distinct rumple in the free corner of the flag. (NASA: S69-38748)

And so NASA designed a telescoping horizontal support that would hinge to the top of the pole. The flag itself was a commercially available nylon flag. A hem was sewn into the top edge into which the horizontal crossbar could be slid. The astronaut deployed the flag by driving the steel-tipped aluminum pole into the surface, then raising the crossbar on its hinge until it locked into the horizontal position. He could then extend the telescoping segment of the crossbar to support the entire width of the flag.

The flag was held oustretched by the crossbar through the top hem. The inner bottom corner was fastened to the pole. The outer bottom corner is free to move. The astute reader will have recognized this as a type of pendulum.

The astronauts said it was hard to drive the pole into the lunar surface. [Ibid.] Apollo 11 had no means of hammering it in. In later missions they reinforced the top of the pole so that a geology hammer could be used to drive it. During the process the flag pole was twisted in the fashion of a drill bit to bore it into the denser layers. Twisting the pole would cause the outer tip of the crossbar to describe an arc with a radius of about five feet (1.5 meters). The free corner of the flag, suspended from the tip, could whip back and forth.

In an atmosphere this motion would be impeded ("damped" in engineering terms) by air resistance. But on the moon there is no resistance from air to the pendulum motion of the flimsy fabric.

This process can be duplicated on earth. Slip the buckle of a belt over one end of a yardstick (or meter stick). Hold the other end of the stick and let the rest of the belt hang underneath it. Now move the stick left and right as if the hand holding it were a pole being twisted. Vary the speed. You will notice some complex pendulum motions at the bottom tip of the belt that look uncannily like the movements of the flag in the Apollo video. Why do we use a heavy object like a belt? Because we need something that won't be greatly affected by the air resistance on earth. In a vacuum the nylon fabric will have some of the same properties as the belt.

In one video you can see the flag move even though no astronaut is touching it. That could only be caused by wind.

In these instances the astronaut has just let go of the flagpole. The flagpole and its horizontal rod are bouncing, resonating in response to the residual motion from the astronaut's manipulation. If the wind is causing this motion then why are the flagpole and horizontal rod moving (bouncing), but the flag itself doesn't move at all? And why, in any of these cases, is there no secondary indication of wind such as blowing insulation on the lunar module or dust raised by the wind.

The flag is off-balance when the pole is perfectly vertical. It is balanced when tilted back slightly. Frequently the crossbar will rotate slightly just after being released by the astronaut, much as the door of an off-balance refrigerator will find its own equilibrium point.

Why didn't the astronauts smooth out the wrinkles in the flag?

They thought it actually looked more familiar to have the flag slightly rumpled. The Apollo 11 astronauts could not get the telescoping crossbar to extend fully. On later missions the astronauts intentionally didn't pull it all the way out.