badmouth vs smirch

badmouth is not a noun while smirch is a noun.

Word	Noun	Adjective	Verb	Adverb
badmouth	No	No	Yes	No
smirch	Yes	No	Yes	No

As verbs, smirch is a hypernym of badmouth; that is, smirch is a word with a broader meaning than badmouth:

badmouth: speak unfavorably about
smirch: charge falsely or with malicious intent; attack the good name and reputation of someone

Other hypernyms of badmouth include asperse, besmirch, calumniate, defame, denigrate, slander, smear, sully.

badmouth (noun)	smirch (noun)
	an act that brings discredit to the person who does it
	a blemish made by dirt

badmouth (verb)	smirch (verb)
speak unfavorably about	charge falsely or with malicious intent; attack the good name and reputation of someone
	smear so as to make dirty or stained