If one has forgotten to groom his nails or did not have an opportunity to cut his nails before Rosh Chodesh and it is quite annoying and unclean, can he/she cut her nails? (not asking about prior to immersion in mikvah)

The custom of not cutting fingernails or getting a haircut on Rosh Chodesh is mentioned in Be’er Heitev Orach Chayim 260, 2. Its source is Tzavaat R’ Yehuda HaChasid. The Aruch HaShulchan (Orach Chayim 260,6) comments that since the custom has no basis in halacha (it is not found in any talmudic source; R’ Yehuda HaChasid warns against it for reasons of safety based on esoteric considerations- Shulchan Aruch HaRav, Orach Chayim 260), there is no special reason to adopt the custom and one is not more demanding in his religious observance for keeping it. If you have hygienic reasons for cutting your nails, you should do so and there is no need to do “hatarat nedarim” beforehand (Shach Yoreh Deah 214 1; Dagul Merevava Yoreh Deah 214).