What do you do to get rid of finger spires?

Every couple weeks, I get these tiny points of skin at the very edge of a fingernail or three but I never know they're there until I scratch my leg and they snag a thread from my socks or I run my fingers through my hair and one strand refuses to let go of my finger.
I get nail clippers and cut them off but, quit often, withing a few hours or over night, they grow back!! Sometimes I'll try filing them down but it either doesn't work they come back before too long.

What do you do to get rid of finger spires?
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