We study the topic of quantum differentiability on quantum Euclidean d-dimensional spaces (otherwise known as Moyal d-spaces), and we find conditions that are necessary and sufficient for the singular values of the quantised differentialdx to have decay O(n −α ) for 0 < α ≤ 1 d . This result is substantially more difficult than the analogous problems for Euclidean space and for quantum d-tori.2000 Mathematics Subject Classification: Primary: 46G05. Secondary: 47L10, 58B34.