It will probably never be proved to everyones satisfaction whether or not they are safe. The health care industry would loose to much money if it proved they weren't. What we do know for sure is that babies don't like it & try to move away from it whenever possible! That should tell you something! Doris Haire has an excellent video explaining the dangers. I can't begin to do it justice here. According to the AMA & ACOG, the possibility of getting sued for not using it, and the complications it can detect if used, outweigh the comfort of the fetus, and the dangers is does pose if used.
I prefer to follow the Hippocratic oath... "above all else, do no harm" better safe than sorry. I avoid using doppler whenever I can hear FHT's with a fetoscope.
BTW... I wonder why more docs don't follow the Hippocratic oath. I also wonder why they seem to change the wording to suit them. Did you realize, it used to say they would promise not to do abortions? That magically disappeared from the wording. It also said the docs would promise to pass on their lifesaving knowledge FREE of charge to all whom desired to learn. Hmmm. wonder what else has changed since Hippocrates wrote it?