The most popular entry here is a warning not to date single mothers. You can see for yourself there are hundreds of comments on it.
Of course, you can avoid single women with kids by avoiding women entirely. That fail-safe solution isn't for everyone, though, especially if your a self-destructive masochist who actually wants to get married.
So how does a guy avoid single women with kids (single mothers), especially since more and more women are mothers the older they get?
Hang out in the pet store by the cat supplies.
But seriously, simply pay attention. Most single mothers won't stop talking about their kids, or will badmouth "the father of my kid(s)". In online dating profiles, their kids will often be in their pictures, even though there really isn't a good reason to show complete strangers pictures of their kids unless they're trying to attract pedophiles.
Any woman who hides the fact she has kids is definitely bad news. If you're just looking for a sex, there's a chance a woman who hides or omits that she has children is also just looking for sex, and so you might think that's a good thing. But ask yourself... how did she become a single mother in the first place? By other men who were just getting together with her for sex? Stay away, unless you have had a vasectomy, and even then, you should probably heed the warning against dating single mothers.
How do you smoke out a single mother who is hiding the fact that she's a mother?
Perhaps by moving things to her place as soon as you can. You don't want her coming back to your place, anyway. If she's reluctant to let you see her place, she might also be thinking about her safety and convenience, or it might be a mess, or she might live with her husband(!), or she might have kids.
It would take a lot of work for her to remove signs of being a mother from around her place. She could try to pass off the other bedrooms as roommates' rooms.
It doesn't matter how things are going, or how attractive you find her. Once you realize she has kids, especially if she hid it, and especially if you haven't had a vasectomy, you need to drop her immediately. Leave. Ghost her.