Word for something that spreads

The term Viral has been co-opted and is no longer considered only a bad connotation. The phrase going viral is generally considered a good thing(unless you are the one being embarrassed in the video or ad)

quickly and widely spread or popularized especially by person-to-person electronic communication

Infectious is my one-word answer.

I'm guessing that it's not absolutely essential to emphasise the "spatial" connotations of spread, though they're often present by implication in my offerings anyway.

I'm also guessing that the thing itself at least in part produces it's own momentum causing it to spread - we're talking more about something like a virus that spreads itself, rather than something like plaster being spread on a wall by a tradesman.

The process of spreading is proliferation. The adjective prolific is normally used of something that has already spread, but if OP doesn't mind sounding a bit "erudite", he can use proliferatory to describe something which has the potential to spread, even if it hasn't actually done so yet.