How should I ask this girl out?

[marvelouslew]

You've answered your own question. With New Year's coming up, invite her to go with you to attend any New Year's events being held in your town. That way it you'll have a chance to express how you want her companionship. And it also gives her an opportunity to tell you about her present "status" as far as dating and new relationships are concerned.

A suggestion - ask her if she might like to attend an early evening "family" type New Year's event. (If she wants her kid to enjoy the holiday and fireworks, etc.) This takes care of any baby-sitting concerns and also lets her know you're sensitive to her situation as a young single mother. Have fun!