Suppose I have a (Haar-random) $m\times m$ unitary matrix $T$, i.e. $TT^\dagger=\mathbb{I}$. Now I take a submatrix from this by taking a subset $(i_1,i_2,\ldots,i_n)$ of $n<m $ rows and a possibly different subset $(k_1,k_2,\ldots,k_n)$ of $n<m$ columns (not a principal submatrix).
Call the resulting $n\times n$ submatrix $U$. What is the probability that $\hbox{det}(U)=0?$ I suspect basically $0$ but I wouldn't know how to prove this.