Consider the Abelian sandpile measure on $\mathbb{Z}^d$, $d\geq 2$, obtained as the $L\rightarrow\infty$ limit of the stationary distribution of the sandpile on $[-L, L]^d$. When adding a grain of sand at the origin, some region topples during stabilization. We prove bounds on the tail behavior of various avalanche characteristics: the probability that a given vertex topples, the radius of the toppled region, and the number of vertices toppled. Our results yield rigorous inequalities for the relevant critical exponents.