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