This was going to be an update to the FAQ page but thought that it would make a good post.
In my experience audible acoustic vibration on a Bass Guitar is almost always due to two things. The most common is Machine head vibration, the other will be an over loosened truss rod. The fender style machine head is constructed in a way that if the 'butterfly' part can become loose without falling apart. I have one bass with this issue and really the only real fix is to replace the head. I've tried rubber bands, sticky-tape, bits of paper and nothing has worked well enough. So, let's fix the thing properly right now!
For the purpose of this post I'm going to go to a little more effort.
Here is an image of the offending part. You can see the remnants of sticky-tap on it.


My first repair attempt was to use a vise grip to apply an incredible amount of pressure in attempt to reduce the movement of the part and stop the vibration. This made a difference but didn't fix the problem completely.


Next, I scuffed up the surface with some sand paper and prepared to solder the two parts together. A heavy duty soldering iron can help here. A little low wattage one is not going to get the surfaces hot enough.
The basics of this operation are that you need to get the surface of the 'job' hot enough to melt the solder.
In the first image here you can see that I've pretty much got the repair done but the solder is a little bubbled and I wasn't confident that repair would hold so I've added heat again, and a bit more solder and completed the job with a bit more patience. When you're finished the solder should shine and not be raised like a bubble.


This should have the job complete. If there is still vibration coming from the part and you can tell by touching it while playing a note, replace the machine head completely.