  The slicing degree of a knot $K$ is defined as the smallest integer $k$ such
that $K$ is $k$-slice in $\#^n \overline{\mathbb{CP}^2}$ for some $n$. In this
paper, we establish bounds for the slicing degrees of knots using Rasmussen's
$s$-invariant, knot Floer homology and singular instanton homology. We compute
the slicing degrees for many small knots (with crossing numbers up to $9$) and
for some families of torus knots.
