Let $A$ be a $m\times n$ real matrix, whose entries are independent, identically distributed random variables, following standard normal distributions (mean zero and unit variance).

What is the distribution of the singular values and singular (left and right) vectors of $A$?

For a symmetric square matrix, the concept of rotational invariance is helpful because (for ensembles where it holds) it allows us to focus on the eigenvalues since the eigenvectors are essentially uniformly distributed.

What is the analogue of "rotational invariance" in the context of a rectangular matrix like $A$? Is it helfpul here too?

Most papers/books I've read on the topic focus on square matrices so if anyone can point out relevant references I'd also appreciate that.