We study the trapping phenomenon of random walks in random environments of i.i.d. random conductances on the bonds of the grid $\mathbb{Z}^d$, the so-called random conductance model. Our main results concern the important model with conductances in $[0, 1]$ and a polynomial-tailed law near zero for which we find the correct order of decay of the anomalous heat kernel for $d \ge 5$. In $d = 4$, the behavior is found to be normal. In addition, we look at the symmetrical situation with conductances in $[1, \infty)$ with a polynomial law at infinity, which also shows opposite return probability behaviors.