I will compare "run into" and "come across" in the meaning of "to meet something or someone". Those verbs have multiple meanings. "Come upon" is marked "literary" or "formal" in modern dictionaries.
run into somebody (rather informal): meet someone you know by chance. The key word here is "someone you know". In this meaning, it doesn't have any negative connotations:
I ran into an old neighbour of mine at the garage this afternoon.
You rarely run into people you know in London.
However, "run into something/someone" in the sense of "to crash (your vehicle) into it/them" is obviously bad.
come across something: to find something unexpectedly when you are not looking for it but are doing something else, as in:
Jill came across her son's diary when she was tidying his room one day.
come across somebody: to meet/find someone by chance, when you are not expecting it, as in:
You must have come across some weird and wonderful people while you were there.
I came across an old school friend of mine when I was travelling in Canada.
I came across a group of children playing.
He's the most unpleasant man I've ever come across.
I came across children sleeping under bridges.
My guess is that you usually run into people you know, whereas you come across people you know or total strangers.
Also notice the following examples where "run into" doesn't sound good or it changes the meaning of a sentence:
Just recently, I came across a familiar looking gorgeous man on facebook.
Was just on a dating website and came across my ex girlfriend on it?
Ever notice how you come across somebody once in a while you shouldn't have fucked with?
Your [sic!] going to come across a lot of shitty bands, and a lot of shitty people.
When you come across your crush, you feel like you are melting and as if you are going to faint.
With things, it's simple. When you find something by chance, you come across it.
The Oxford Phrasal Verbs Dictionary marks "come across sb/sth" as very common. It doesn't mark "run into" as common.
Sources: the Longman Phrasal Verbs Dictionary, The Oxford Phrasal Verbs Dictionary, the Cambridge Phrasal Verbs Dictionary, the Longman Language Activator, The Oxford Learner's Thesaurus