Over the years I've witnessed a variety of incidents where bystanders did nothing -- however once someone makes the first move, it seems to become easier for others to lend a hand.
When I was a teen, I dated someone who was a red cross volunteer, which prompted me to take a couple of first aid classes. This has easily been the most useful subject I've ever studied.
I've been witness to a suicide attempt, a cardiac arrest in a bar, an elderly lady who fell down the stairs at the opera, several cases of guys beating on their wives in public, two drunken attempted robberies, a variety of (literal) gay bashing, random subway violence, and oodles of shit I can't recall. The latest was two weeks ago: a hit and run with a teen victim with a very nasty headwound.
In every one of those cases, no one did anything. The old lady at the opera was the worst, I hadn't seen it happen, but I noticed a crowd of people on the staircase on my way out. Maybe 20 people standing around her in a circle. I asked if anyone had called for an ambulance yet, and no one even gave a coherent response...
I've also been the victim of a random assault, in a very busy park in broad daylight, and no one helped out. No one even dialed the emergency number, no one even approached me to ask whether I was ok afterwards. There were probably around a hundred witnesses.
I don't know whether this is a regional problem; most of these incidents occurred in Belgium. Belgians aren't the most outgoing or the most eager to take risks in general, but still.
Don't count on others to make the first move. Learn first aid and help out whenever someone is in trouble. Even if it does mean you'll be late for that important meeting or you could get blood on your shirt.
Violence is a bit trickier; what's worked well in my experience is to walk up to the aggressor, but to pick someone else from the crowd and plainly tell them "give me a hand". That gets the ball rolling.
Edit/addendum: every time something like this happens, I get an adrenaline rush and am scared out of my mind. I'm sure this happens to the other witnesses as well. I get it, it's hard to push through that, but do it anyway.