How do you do this? I know it's possible and I sorta remember it needing "Run AI Script" actions but searching for info on this hasn't turned up anything.

As far as i know, you cant control spider mines. Maybe you're thinking of interceptors and scarabs?

http://www.staredit.net/starcraft/Ordering_and_using_interceptors_and_scarabs

It's not possible.

You could probably simulate it using an unreachable target or something... but never directly do it.

Yeah I think that's how it was done, some kind of unreachable/fleeing target. Any ideas on how to abuse that then? I'm pretty sure that making a unit that the spidermine acquires, then moving the unit away will just make the spidermine forget about it instead of following it.

Moving the unit makes the Spider Mine lose its target and burrow again (even if you move the unit to where it already is). The Spider Mine can outrun a speedling, and they don't target Vultures, so you won't be able to make it "chase" a unit unless you slow down its movement.

Purely for the sake of "it's possible," I present:

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That's really cool, so you can control a spidermine for a scripted sequence at least, but what's the zergling sprite in the bottom right for?

The Zergling is what the Spider Mine is chasing. I just used a Sprite/DoodadState trick to make it virtually invisible. If you look at the "Hide Zergling" trigger, you'll see I'm enabling the sprite to remove its collision, disabling/enabling it to cloak it, giving vision to Player 1 and back to make it show as cloaked, and letting it burrow to remove its shadow. You can remove the give actions in that trigger to see the Zergling running away from the mine.