"What's a nice girl like you doing in a place like this?"
Just kidding. It would be rather awkward to hit on someone in church, I think, and I have to admit I don't think it's really appropriate to talk until after Mass. However, most people bolt as soon as they can, so you might have to be quick. You could smile and say "Hi" and if she doesn't bolt continue the conversation from there, possibly. But it would be a lot easier to introduce yourself at coffee after Mass (if your church has it and she goes to it) or at practically any other church event.
Been there, done that. Three different occasions I started a little conversation, to which he (same man) responded in a friendly and courteous manner. Even ran into him at a coffee shop and he was pleasant.
But it didn't do me a bit of good in the long run....
OTOH, I have a friend who was approached by a lady after Mass, and it resulted in a two-year relationship.
I would never ask a man anything after Mass. That is totally up to him if he is interested in me. Unfortunately, those who have started to talk to me have been rather creepy.
I think the best way would be to invite a woman to after church activity when he is in a group. I would feel more comfortable that way.
Plus you will probably see a married woman with her husband if you see her often in church. It's unusual for a woman to go to Mass with a man unless they have a romantic interest in each other (or is her brother or cousin maybe, which you can probably notice if you look at how they interact non-verbally).