In some states, like Utah, it is automatically assumed that the husband is the father. A DNA test or the husband signing paperwork denying paternity are needed to prevent that.
Lady:"Hi, I'm having a baby! Can I get pre-admitted please?"
Hospital admissions: "Ok, let's collect some information"
[lots of questions, typing, etc]
Hospital admissions: "Are you married?"
Lady:"..No."
Hospital admissions: "Okey Dokey. That about covers it"
If the hospital doesn't know, how would they ever know to put the husband on the Birth Certificate? Just make sure you're the one there when they're filing all that paperwork and I'm pretty sure you're gonna be fine.