Thom--Boardman strata $\Sigma^I$ are fundamental tools
in studying singularities of maps. The Zariski
closures of the strata $\Sigma^I$ are components
of the set of zeros of the ideals $\Delta^I$ defined
by B. Morin using iterated jacobian extensions
in his paper `Calcul jacobien' ({\em Ann. Sci.
\'Ecole Norm. Sup.} 8 (1975) 1--98). In this paper,
we consider the question of when
the 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.
