You frequently encounter this one online, but in real life they're not an issue for most of us once we're out of our teens.
This type of cyclist hater (Type 1) is typically young, most often female, and desperate to fit in. They want to be accepted by their friends and peers as 'normal', and they will target cyclists as easy prey. Rather like the clique of teenage girls picking on the outsider, they see deprecating cyclists as a way of reinforcing their own status within their clique.
This type of cyclist hater is amazingly common. You'll find them on twitter, you'll find them on youtube, on facebook, in fact you'll come across them anywhere online that folk gather online to mouth off nonsense at each other. And while these folk are little threat on the road (after all, few of them are old enough to drive, and they're mostly more bark than bite) the constant pitter patter of their anti-cyclist chorus has the effect of keeping cyclist hatred normal - for as long as they go unchallenged and unashamed for what amounts to nothing more than a bland prejudice, hating cyclists (and the occasional extreme act that go alongside that) will be considered socially acceptable.
I know, this sounds far fetched. I grant you I'm speaking highly speculatively so far. But I regret to say that the net is full of examples of the Type 1 cyclist hater. I'm not overstating it when I say they're everywhere - it is absolutely, totally normal to pour scorn or outright hatred on to cyclists. You don't need a pretext, you don't need a reason, in fact its increasingly used as a neutral means for starting a conversation. It is considered normal and reasonable to hate cyclists, its respectable, at least among these folk.
You'll note looking though examples how common other prejudices are too - homophobia is common among them, largely for the same reasons they abuse cyclists. They see it as safe, as easy. They are, in short, typical bullies.
How should we respond to them? Well, for the most part, that depends on whether you think they're worth it. Its hard not to respond to the worst variety of braindead idiots who think its amusing to threaten murder, but remember if these barely pubescent creatures had the courage of their convictions they'd already be in jail. Generally speaking if you're going to respond its best to do so in a reasonable, detached sort of way; express incredulity or amazement that they mean what they're saying, perhaps ask them why they think its okay to stereotype about cyclists, and what other groups they feel they can reasonably stereotype and denigrate in this way. If that doesn't work then publicise them and allow others to pour scorn on them too; the one thing these kids are looking for, the overwhelming urge they have, is to be accepted as normal, to be part of a pack, and when the 'pack' turns on them they'll retreat back in shame.
I feel that its sometimes worth responding to them, sometimes not. But it tends to be the case that once three or four people have responded these kids back off and realise that what they're saying really is offensive. I rather hope that in nipping this off at the bud we can prevent them turning into the alltogether more dangerous tpe 2 cyclist hater...