Praise be to Allah.
Clipping the nails is one of the Sunan al-Fitrah as indicated by authentic hadiths, and it is mustahabb to start on the right, because it is proven in Sahih al-Bukhari (163) that ‘Aishah (may Allah be pleased with her) said: “The Prophet (peace and blessings of Allah be upon him) used to like to start on the right when putting on shoes, combing his hair, purifying himself and in all his affairs.”
Al-Nawawi (may Allah have mercy on him) said in al-Majmu’ (1/339): “There is scholarly consensus that clipping the nails is Sunnah, and that applies equally to men and women, and to the hands and feet. It is mustahabb to start with the right hand, then the left, then the right foot then the left.”
And he (may Allah have mercy on him) said in Sharh Sahih Muslim (3/149):
“It is mustahabb to start with the hands before the feet, and to start with the forefinger of the right hand, then the middle finger, then the ring finger, then the pinkie finger, then the thumbs, then to move to the left hand and start with the pinkie finger then the ring finger, until all the fingers are done, then to move to the right foot and start with the smallest toe on the right foot and end with the smallest toe on the left foot. And Allah knows best.”
The Hanbalis mentioned a different sequence of fingers when clipping the nails, but no sahih report has been narrated from the Prophet (peace and blessings of Allah be upon him) concerning this matter.
Al-'Iraqi (may Allah have mercy on him) said in Tarh al-Tathrib (2/77): “There is no proven hadith which may be followed with regard to how to clip the nails.”
Ibn Hajar said in Fath al-Bari (10/345):
“There is no proven hadith about the order of fingers when cutting the nails. … Ibn Daqiq al-’Eid rejected the order mentioned by al-Ghazali and those who followed him, and he said: There is no basis for any of that, and to suggest that it is mustahabb when there is no evidence for that is abhorrent in my view. If anyone were to suggest that it is mustahabb to start with the right forefinger because of its honourable status, there is no basis for the rest of this sequence. Yes there is a basis for starting with the right hand and the right foot, which is “he liked to start on the right.”
For more about sunan al-fitrah, please see this answer: 75727
And Allah knows best.