It is quite possible that the heter meah rabbonim occurs far more often, but it tends to get very little publicity. It is a process that involves a sense of victory and shame at the same time. On one hand, the divorcing husband manages to win the get war, and slips out of the bond that stopped him from remarrying. But it is also a nuclear option, and makes the husband appear to be a mean, narcissistic person.
I have actually see a few of these. I would NEVER allow my child to consider marrying someone who received his heter to marry by the heter meah. Firstly, (I am not a posek and am not qualified to establish halacha) I frown on the way the signatures are obtained, with very few of the signatories knowing anything at all about the case. In cases I observed, the document was carried into a kollel, and made its rounds collecting signatures of yungerleit who were uninformed about any of the facts of the case. I would not consider my signature cheap, and struggle when someone else does. Secondly, there are opposite extremes of situations that result in seeking heter meah. One extreme is a wife that refuses to accept a get out of stubbornness, such as impossible demands. The opposite end is the husband who refuses to give the wife anything, and since the affairs cannot be settled, she will not accept the get. This husband then turns around and seeks the heter meah rabbonim. These situations are extremes, and their are always unique variables to a specific case. But all too often, the situation of heter meah stinks.
It may be more common than we think. Just lots of cases stay below the radar.
For the most part, heter meah rabbonim is used as a threat to leverage some power to the husband.