> I've inspected the [white things] on my trampoline (where they're easy to see, because the tramp's black) and I have found them to be poor little insects of many types, some dead, others feebly crawling about, with many soft white threads bursting out of their abdomen.
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> I'm guessing that it is some sort of predatory fungus that eats the bug's insides, but I'm not sure. Can anybody help me identify what is causing this wanton slaughter?

You're absolutely right. I don't know a lot about fungi, but there are certain parasitic funguses. I can't think of one which creates long, visible tendrils when the insect is still alive, however, there is at least one species which lands on an insect, and taps into its various bodily systems/matter and derives its energy to grow from such. Eventually nothing is left but a long white stalk growing from an empty husk.

Rabbit"eewww, where do I learn these things?!?!?"lord