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What materials are better cut with the compression type cut of clippers, vs the slicing of a knife or the mechanical separation of scissors?

Well, the hangnail issue is close. I've used my nail clippers to trim the excess skin from busted blisters and dead skin from healing wounds. Because of the arc, they're not very precise, but sometimes it's all I've got.

Trimming off hangnails and broken nails is not just a comfort issue in some situations, by the way. Long term discomfort can go well beyond distraction.