We study the criticality and subcriticality of powers $(-Δ)^α$ with $α>0$ of the discrete Laplacian $-Δ$ acting on $\ell^2(\mathbb{N})$. We prove that these positive powers of the Laplacian are critical if and only if $α\ge 3/2$. We complement our analysis with Hardy type inequalities for $(-Δ)^α$ in the subcritical regimes $α\in (0,3/2)$. As an illustration of the critical case, we describe the negative eigenvalues emerging by coupling the discrete bilaplacian with an arbitrarily small localized potential.