I've three moments, all involving my children:
1) My daughter's childbirth. My partner (now wife) had pre-eclampsia and needed to be induced at 35 weeks. The induction worked quickly and was going well, but all of a sudden Evie's (my daughter) heartbeat wasn't coming back as quick as a should after contraction. They whipped my partner into surgery to get Evie out, by any means, which was a panic inducing enough, but then my partner started looking decidedly ill too. I just sat next to her sobbing with a whirl of people around and beeps going off.
Thankfully, Evie (though only 4lb 9oz) was absolutely perfect and so was mum once baby was out. They were home after 3 days.
2) We were around my parents for my sister's birthday. Evie was about 2 or 3 and was outside on the trampoline, being watched by my dad and I from the conservatory. She just collapsed. I ran outside and just grabbed her up (I thought she hadn't banged her head or neck, but I just operated on instinct). We rushed inside, I got her on the floor and she was going blue and wasn't breathing. Mum was on 999 but my partner and my dad were just sat helpless watching. I think what made it worse was my dad couldn't help like you just expect your parents to do - like the realisation of a dad being just a normal person.
I gave my daughter mouth to mouth, which cleared her airways and she started coming around. Ambulance arrived and she started to come around more fully. Turns out she had a fit, high temp etc and is apparently quite common. By the time she got to hospital, it was as if nothing had happened (and after she puked on me!

) I thought I was watching my little girl die in front of me.
Again, she's absolutely fine now.
3) More recently (this August) my wife was induced with our second. She was 37 weeks, baby was a good size and all appeared well. Laura was induced between 3-5pm and by 7 the contractions were really kicking in. We got into the suite (which just so happened to be the same one Evie was
almost delivered in) and things happened fast. By 9.30pm the room was full of people, Noah's heartbeat was dropping like Evie's had except Laura hadn't had an epidural and he was much bigger. The doctor basically yanked him out and all I could see was a dark blue face of my son, thinking he was dead. So much panic, apart from the paediatrician. She just whisked him onto the birthing station, got him breathing, then whisked him away to Neo-natal.
We got to see him at 11pm. Stable, but very weak. He was 9lbs, and all the others in there were tiny. He was so big it didn't seem right. He was without oxygen for almost 8 mins (that they can observe, possibly less). We were told he needed a brain scan, his shoulder was potentially damaged (irreparably so if nerves). We eventually left at 1pm, my wife back to a ward and me to home.
I got up the next morning, god knows how I slept, had a shower and just sobbed. Cleaned myself up, got myself to hospital and just sat with him crying. The beeping of all the machines set your heart racing which made it worse. The nurses and doctors in there were immense however - so caring, reassuring and hardworking.
He was taking well to all the treatments, drips, etc. and eventually went down to Laura on the ward. We still had the brain scan hanging over us, which kept getting delayed. We were surprised when a doctor came in with a lovely big pram and asked us to come for the scan. We just sat there shaking as they did the ultrasound and as he scanned each part of the brain for bleeds or damage - if I could be sick I think I would be (that's another story!). As each section came back clear I could feel the tension lift and my wife's hand loosen.
His shoulder had also been for x-ray and wasn't broken and after physio and sessions with a consultant that's fine too. He's now a greedy little 5 month old with what seems like a constant smile.
I can't rank which has made me most scared, probably #1 purely for the fact I completely lost it.
EDIT: Wow, that was a long post for me. No-one will read it now, but at least it was cathartic to write it!