Morpheme-based morphology comes in two flavours, one Bloomfieldian and one Hockettian.
For Bloomfield, the morpheme was the minimal form with meaning, but it was not meaning itself.
|Level||Lexical form||Grammatical form|
|minimal signalling units||phonetic form||tactic form|
|form without meaning||phoneme||taxemes|
|form with meaning||morpheme||tagmeme|
For Hockett, morphemes are meaning elements, not form elements. For him, there is a morpheme plural, with the allomorphs -s, -en, -ren etc.
Within much morpheme-based morphological theory, these two views are mixed in unsystematic ways, so that a writer may talk about "the morpheme plural" and "the morpheme -s" in the same sentence, although these are different things.