sign up vs fall in

sign up and fall in both are verbs.

Word	Noun	Adjective	Verb	Adverb
sign up	No	No	Yes	No
fall in	No	No	Yes	No

As verbs, fall in is a hypernym of sign up; that is, fall in is a word with a broader meaning than sign up:

sign up: join a club, an activity, etc. with the intention to join or participate
fall in: become part of; become a member of a group or organization

Other hypernyms of sign up include get together, join.

sign up (verb)	fall in (verb)
join a club, an activity, etc. with the intention to join or participate	break down, literally or metaphorically
engage by written agreement	become part of; become a member of a group or organization
	to take one's place in a military formation or line