stick in vs lay

stick in is not a noun while lay is a noun.

Word	Noun	Adjective	Verb	Adverb
stick in	No	No	Yes	No
lay	Yes	Yes	Yes	No

As verbs, lay is a hypernym of stick in; that is, lay is a word with a broader meaning than stick in:

Other hypernyms of stick in include place, pose, position, put, set.

stick in (noun)	lay (noun)
	a narrative poem of popular origin
	a narrative song with a recurrent refrain

stick in (adjective)	lay (adjective)
	not of or from a profession
	characteristic of those who are not members of the clergy

stick in (verb)	lay (verb)
place, fit, or thrust (something) into another thing	put in a horizontal position
insert casually	put into a certain place or abstract location
	lay eggs
	prepare or position for action or operation
	impose as a duty, burden, or punishment