I had one with each child. About 12 years ago there was talk about possible damage to fine hairs (maybe implicated in ear infections?). My older son was checked for position a few days before delivery, and my youngest was done right away for date confirmation.
I did get to be in a study about fetal monitors (I think the results were that there were more interventions on mothers with monitors, and I think the upshot was there were probably more unnecessary interventions).
I had an internal fetal monitor with my first delivery. Completely aside from the fact that the dr had to squeeze his hand in unspeakable places to attach the line to my undelivered baby's head (which I occasionally complain about 18 years later), being flat on my back in bed was unconducive to my labor, I think. I'm not surprised by that result.
Yup, I got the internal monitor too since my water had broken earlier and I was considered high risk at the time. It was kind of cool to watch the monitor, but they ended up doing a forceps delivery because of deceleration in the heartbeat (if I recall—its been 32 years!) which I think later they decided was normal during labor. But I may misremember parts of it. They also did a little test on my son when he was about 9 months old (not an IQ test exactly, but to see where he was developmentally). Of course, he was a genius! ;)