I can think of no known instance in Greek mythology when the Fates (Greek Moirai) used an instrument to cut the thread. We hear a lot about the weaving of fate at the loom, but virtually nothing about the cutting. Cutting implements of various sorts do show up in artistic depictions of the fates (but I'm having trouble finding an example right now in which they are clearly holding shears of any sort; mostly she holds a spindle). Certainly the shears had no name in antiquity. Even very late Hyginus mentions no named shears.

The literary sources for this are Homer (who usually mentions only one Moira), Hesiod, Aeschylus, maybe Aristophanes, Plato's Republic, and probably Ovid. The excellent Theoi site has a good run-down of the Moirai here, which you could check (scroll down).