What was the caste of Vidur's wife? To which caste does Vidur's wife belong?
The caste of Vidura's wife is Ugra, which needs to be treated as Sudra.
Vidura is a sudra. You can observe it from this answer
"Vidura said, 'I am born in the Sudra order and, therefore, do not venture to say more than what I have already said. The understanding, however, of that Rishi leading a life of celibacy, is regarded by me to be infinite. He that is a Brahmana by birth, by discoursing on even the profoundest mysteries, never incureth the censure of the gods. It is for this alone that I do not discourse to thee, upon the subject.'
Vidura married a woman born to a Kshatriya father and a Sudra mother. You can observe it from this answer.
"Meanwhile the son of the ocean-going Ganga heard that king Devaka had a daughter endued with youth and beauty and begotten upon a Sudra wife. Bringing her from her father's abode, Bhishma married her to Vidura of great wisdom. And Vidura begot upon her many children like unto himself in accomplishments.'"
So, the caste of the wife of Vidura is Ugra.
A Kshatriya may take three wives. In two of them (viz., the one taken from his own order and the other that is taken from the order immediately below), he takes birth himself (so that those children are invested with the status of his own order). His third wife being of the Sudra order is regarded as very inferior. The son that he begets upon her comes to be called as an Ugra.
But people born of Vaisya women by Ksatriyas and Brahmanas should be treated as Shudras, also men born of Shdra women.
Thus, the wife of Vidura is of the Ugra caste and needs to be treated as Sudra.