I've heard in a lecture from a reliable teacher on the subject of slow shutter speeds that 1/15 is the worst shutter speed to work with because the second the mirror opens and flaps up the vibration starts at exactly 1/15 of a second (on a 35mm camera anyway). He recommended either go one down on the aperature and shoot at 1/30 or increase the aperature a stop and shoot at 1/8 as the photo will suffer less proportional vibration. The logic is that, even though the immediate vibration may be the same, it's taking in less light in the initial vibration from the mirror slap and bringing in more light when the vibrations stop.
Is this correct? I've never actually tried it before (I just muscle out the tripod), but I remember taking really good concert photos at 1/8 of a second with no flash as long as I could sturdy the camera. I'm a little skeptical of the logic, but it does make sense in a sense.
The logic is correct. Sort of.
Assuming the vibrations start at 1/15 sec. after the mirror went up, you would also need to know how long the vibrations last and when the exposure starts to know how much the mirror induced shake will affect the picture.
But supposing the exposure starts exactly then too, and that the shake takes 1/15 to die down, the longer the exposure, the smaller the proportion of it that is affected.
Shorter exposure will reduce shake only if they start (and possibly/ideally finish) before the shake starts. But given that the mirror is hitting the frame (the thing that is supposed to cause the shake) before the exposure can begin, how big is the chance of that happening?
It would depend on the speed with which the movement propagates through the camera, i.e. the camera not being quite ready to shake yet, but quite willing and able to complete the rest of the release cycle.
What would be not correct is that you can name one single speed as generally the worst one. That would be different for different cameras.
And what would also not be correct is any assumption that mirror induced shake is something we need to worry about when we are holding the camera in our hands...
If present, it is orders of magnitude smaller than hand induced shake.
So unless you have your camera on a tripod, do not worry about the mirror!