Are we allowed to clip our nails during tzom gedalia?
It is permitted to cut nails on the Tzom Gedalia, etc. The reason is that halachically it is permitted even to shower etc. on these fast days, however it is preferable not to take a hot shower on these days. Cutting nails however is questionable if it is even assur during the nine days. Therefore we are not going to take this hanhagah to this extent.
See O:CH 550-2, 551-3, M:B 551- 20, Shar Hatzion 550-8, Aruch Hashulchan 550-3, 551- 32 that rules that they may be cut, Poskim.