Praise be to Allah.
Islam tells us to trim our nails, because this is one of the five qualities dictated by the fitrah (natural inclination of man). If one throws them in the garbage or buries them, there is nothing wrong with doing either (Fataawa al-Lajnah al-Daa’imah, 5/174). But if one is afraid that they may fall into the hands of a practitioner of sihr (witchcraft), then he should bury them or throw them somewhere where such people will not be able to get their hands on them. And Allaah knows best.