To what extent is the Singular Value Decomposition unique?

For distinct singular values, SVD is unique up to permutations of columns of the $U,V$ matrices. Usually one asks for the singular values to appear in decreasing order on the main diagonal so that uniqueness is up to permutations of singular vectors with the same singular values.

When singular values are repeated, you have additional freedom of rotating their subspace by an orthogonal matrix $O$, e.g. $U[:,i:j]O$ and $V[:,i:j]O$ for the subset of columns $i:j$ which correspond to the same singular value.