Sings the praises of mystery woman in lace stole (6)

I believe the answer is:

extols

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'sings the praises of' is the definition.
(I've seen this before)

I can't explain the remainder of the clue.

(Other definitions for extols that I've seen before include "Sings the praises of" , "Enthusiastically praises" , "'Praises, lauds (6)'" , "Praises highly, lauds" .)

I've seen this clue in the Irish Times (Crosaire) and the Irish Times.

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