I dated several single moms, and wound up marrying one of them. Still got her, somewhere around here. 
First of all, there is a “window” of women I will date. I wasn’t gonna date all women, yes? ALL guys have a “window;” can’t be too old, too young, too beautiful, too ugly, too fat, too skinny, whatever.
Now, for some guys, “single mom” falls outside that window. It did not, for me; I was okay with it, if the woman was interesting, but other guys just flat would not do it.
As to whether I would continue with a dating relationship or get serious, that largely depended on the woman in question. I dropped one woman because she kept trying to get her kids involved in “us,” less than a month after we’d first met. It felt clingy and creepy, like she was trying to shoehorn me into a husband-and-daddy-shaped hole in her life.
This other woman I dated later made it clear that something could happen, but that her child was all she had, and that she wasn’t gonna shortchange her kid just to suit me.
That made sense, and we were able to work with it. Her kid disliked me on sight, but later thawed after I brought Mom flowers… and Daughter a bouquet of bubble gum and small toys. Got along fine after that. Been getting along peachy for more than a decade now.
'Course, you got to find the right GUY, too…