I know how to classify all of them (it's a nice, not so easy exercise), and I guess this was investigated long ago, but haven't been able to find a reference. Using keyword searches leads to similar results but not to the one I want.
You can refer Lallements book on Semigroups
The new book "Theory of Regular Semigroup" a 2018 publication by K.S.S.Nambooripad will be a very much useful
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