  In this paper, we study reducible surgeries on knots in $S^3$. We develop
thickness bounds for L-space knots that admit reducible surgeries, and lower
bounds on the slice genus for general knots that admit reducible surgeries. The
L-space knot thickness bounds allow us to finish off the verification of the
Cabling Conjecture for thin knots, which was mostly worked out in
\cite{DeY21b}. We also provide a new upper bound on reducing slopes for
fibered, hyperbolic slice knots and on multiple reducing slopes for slice
knots. Our techniques involve the $d$-invariants and mapping cone formula from
Heegaard Floer homology.
