Your opinion on dopplers and ultrasound?

I just got in an argument with my mother over whether fetal dopplers are safe. She said that she only ever had one with me. This is my first pregnancy and so far I have had to have several due to cramping and fluid problems. I'm only 26 weeks and have also had to have several non stress tests which use doppler ultrasound to hear the heart beat. I also have a fetal doppler I use at home every now and then. I was just wondering. Are fetal dopplers and ultrasounds safe? I've heard from my doctor that they are but they mom seems to think my baby will come out 100% deaf or mentally challenged and unhealthy.