Prickliness is a characteristic of being sharp and spiky, either literally or figuratively. When you start pruning that rose bush, don't be surprised by the prickliness of those thorns.
Use this noun when you're talking about thorny things, like brambles and burrs. You can also use it figuratively when someone is very quick to be offended or irritated: "I don't want to proofread your book report — your prickliness means you'll get mad at me no matter what I say!" This has been a common way to use prickliness since the late 1800s.