According to Halacha, is one allowed to file their nails with a nail file on Shabbat? Would this fit into a specific melacha that is prohibited? Source for this would be appreciated
Refer to Rabbi Dovid Ribiat's The 39 Melochos, Vol. III p.680
Cutting or removing fingernails or toenails with an instrument (even a knife) is Gozez M'deoraisa. Similarly, filing one's nails is also Gozez M'deoraisa because the tiny shavings are removed with an instrument.
The Shemiras Shabbos Kehilchasa, also says like this albeit without a specified reason. (See the Feldheim English Edition, Vol. I, 14:54, p.162)
On Shabbath and Yom Tov one is not allowed to cut, trim or file nails, whether with scissors, a nail file or any other instrument or by biting them.