We prove that the Abelian sandpile model on a random binary and binomial tree, as introduced in \cite{rrs}, is not critical for all branching probabilities $p<1$; by estimating the tail of the annealed survival time of a random walk on the binary tree with randomly placed traps, we obtain some more information about the exponential tail of the avalanche radius. Next we study the sandpile model on $\mathbb{Z}^d$ with some additional dissipative sites: we provide examples and sufficient conditions for non-criticality; we also make a connection with the parabolic Anderson model. Finally we initiate the study of the sandpile model with both sources and sinks and give a sufficient condition for non-criticality in the presence of a finite number of sources, using a connection with the homogeneous pinning model.