Anybody building a site in that requires users to create accounts is going to face this language challenge. You'll probably have this language strewed across your entire site, from prominent calls-to-action in your homepage hero, to persistent header buttons, to your documentation.
So which is correct? "Sign Up" or "Signup"? Let's try to figure it out.
With some light internet grammar research, the term "sign up" is a verbal phrase. As in, "sign" is a verb (describes an action) and "sign up" is a verb plus a complement — participial phrase, best I can tell. That sounds about right to me.
My best guess before looking into this was that "signup" isn't even a word at all, and more of a lazy internet mistake. Just like "frontend" isn't a word. It's either "front-end" (a compound adjective as in a front-end developer), or "front end" (as in, "Your job is to work on the front end.").
I was wrong, though. "Signup" is a noun. Like a thing. As in, "Go up the hallway past the water fountain and you'll see the signup on the wall." Which could certainly be a digital thing as well. Seems to me it wouldn't be wrong to call a form that collects a user's name and email address a "signup form."
"Sign-up" is almost definitely wrong, as it's not a compound word or compound adjective.
The fact that both "sign up" and "signup" are both legit words/phrases makes this a little tricky. Having a verbal phrase as a button seems like a solid choice, but I wouldn't call it wrong to have a button that said "Signup" since the button presumably links directly to a form in which you can sign up and that's the correct noun for it.
Let's see what some popular websites do.
Twitter goes with "Sign Up" and "Log in." We haven't even talked about the difference between "Log in" and "Login," but the difference is very much the same. Verbal phrase vs. noun. The only thing weird about Twitter's approach here is the capitalization of "Up" and the lowercase "in." Twitter seems giant enough that they must have thought of this and decided this intentionally, so I'd love to understand why because it looks like a mistake to my eyes.
Facebook, like Twitter, goes with "Sign Up" and "Log In."
Google goes with "Sign in" and "Create account." It's not terribly rare to see companies use the "Create" verb. Visiting Microsoft's Azure site, they used the copy "Create your account today" complemented with a "Start free" button. Slack uses "Sign in" and "Get Started."
I can see the appeal of going with symmetry. Zoom uses "SIGN IN" and "SIGN UP" with the use of all-caps giving a pass on having to decide which words are capitalized.
Figma goes the "Sign In" and "Sign up" route, almost having symmetry — but what's up with the mismatched capitalization? I thought, if anything, they'd go with a lowercase "i" because the uppercase "I" can look like a lowercase "L" and maybe that's slightly weird.
At CodePen, we rock the "Sign Up" and "Log In" and try to be super consistent through the entire site using those two phrases.
If you're looking for a conclusion here, I'd say that it probably doesn't matter all that much. There are so many variations out there that people are probably used to it and you aren't losing customers over it. It's not like many will know the literal definition of "Signup." I personally like active verb phrases — like "Sign Up," "Log In," or "Sign In" — with no particular preference for capitalization.