Thom--Boardman strata $\Sigma^I$ are fundamental tools in studying singularities of maps. The Zariskiclosures of thestrata $\Sigma^I$ are componentsof the set of zeros of the ideals $\Delta^I$ defined by B. Morin using iterated jacobian extensionsin his paper `Calcul jacobien' ({\em Ann. Sci.\'Ecole Norm. Sup.} 8 (1975) 1--98). In this paper,we consider the question of whenthe Morin ideals $\Delta^I$ define Cohen--Macaulay spaces.We determine all $I=(i_1,...,i_k)$ such that $\Delta^I$defines a Cohen--Macaulay space alongthe $\Sigma^{i_1}$ stratum. 1991 Mathematics Subject Classification: 13D25, 14B05, 14M12, 58C25.
