We classify all the cocyclic Butson Hadamard matrices BH(n, p) of order n over the pth roots of unity for an odd prime p and n p ≤ 100. That is, we compile a list of matrices such that any cocyclic BH(n, p) for these n, p is equivalent to exactly one element in the list. Our approach encompasses non-existence results and computational machinery for Butson and generalized Hadamard matrices that are of independent interest.