I never cut them on Motzai Shabbos because I feel like it’s rude towards Shabbos but there is no Halachic source for that.
The Gemara says that a pregnant woman who walks over clippings can lose her baby. We assume that this is for mystical reasons, but I was once given an excellent practical Pshat. Someone pointed out to me that it was entirely possible that a pregnant woman who stepped on a nail felt like she had just been bitten by a scorpion and lost her baby due to the trauma/fear.