Consider a matrix $\mathbf{F} \in \mathbb{K}^{m \times n}$ of univariate polynomials over a field~$\mathbb{K}$. We study the problem of computing the column rank profile of $\mathbf{F}$. To this end we first give an algorithm which improves the minimal kernel basis algorithm of Zhou, Labahn, and Storjohann (Proceedings ISSAC 2012). We then provide a second algorithm which computes the column rank profile of $\mathbf{F}$ with a rank-sensitive complexity of $O\tilde{~}(r^{\omega-2} n (m+D))$ operations in $\mathbb{K}$. Here, $D$ is the sum of row degrees of $F$, $\omega$ is the exponent of matrix multiplication, and $O\tilde{~}(\cdot)$ hides logarithmic factors.