The aim of this paper is to study the set LSK(X) of the compactifications of X which can be obtained as a supremum of singular compactifications. We prove that a compactification aX of X belongs to LSK(X) if and only if aX is the supremum of the set SC_a of the singular compactifications induced by the maps from X to a compact subspace of R which extends to aX. Then we are able to characterize the compactifications of a given space X which belong to LSK(X) and the spaces for which every compactification is the supremum of singular compactifications. Finally we prove that if SC_a generates aX then aX belongs to LSK(X) and we give some sufficient conditions in order to SC_a generate aX.