I'd bet the same would apply to most US too if you wouldn't purpusefully dress like a "rich runner"... (or go the other extreme and end up looking like a home robber on the run form the police :P).
Women running are at risk for sexual assault, but this is less correlated with "bad neighborhoods".
Or access to a treadmill.
;-)
I (a man) feel uncomfortable walking, let alone running, in any neighborhood, safe or not. A man walking/running in a more upper-middle class neighborhood (or manicured gated community) always attracts attention and someone's bound to call the cops on you "just to be on the safe side" :) Safer to whiz by in your car and lift weights in your garage.
I’ve been stopped/checked out by police while walking with friends late at night in small towns, but running (in running clothes) seems to put one beyond suspicion.
Like, I can't even fathom a situation where I would be worried about walking somewhere just in case someone calls the police on me. That's crazy.
More concretely: are there places in big cities in the USA where you should be very cautious walking around at night? Absolutely. But the exact same thing applies in London and Manchester too. Robbery is an issue there too.
To some extent, people in the USA just make a bigger deal out of it. You'll also find that when Americans discuss travel abroad they're also very worried about safety. IMO it's a cultural mentality more than the specific realities of specific places.