Its actually the clam's flesh, its foot.

A clean thin edge is the key I've found. Sharpness (for the uncoordinated) is not a great added factor for fear of either harming the clam, or yourself, or something else.

The clams byssal gland makes thin threads which it uses to hold itself in place. Cutting the threads does not hurt the clam in anyway. It is sort of like getting a hair cut. It does not hurt unless it's a bad one lol. I always use a razor blade to cut instead of scraping the threads as you do not want to put excessive force on the byssal gland. The byssal gland also has a 1/4 walnut shaped thing that it holds inside which the threads attach to. Once this gets too many cut threads the clam will expel it and start over again. People sometimes mistake it for the byssal gland and think their clam is dieing.

Cheers,
Tim