I read this question on a forum, and had to share my personal favorite approach.
Talking to a girl in the supermarket was probably the biggest obstacle to me learning how to be social and flirty everywhere. Maybe it's the bright lights, the commerce, the potential for boyfriends to turn the corner and spot catch you mid-grope, whatever. But it's actually fun when you learn how to enjoy it. Here's how I get my shop on.
Supermarket is a super-normal situation. Act super-normal. Forget the stuff designed for clubs. Just be friendly, flirty, and show you have some manners.
Look at what she is buying. Make a comment on it. Anything.
Whatever you do, just have fun with it. Think less, do more.
"Wow, you got a lot of good food there. I usually just buy beer and a pack of Twizzlers."
"Yes, well..."
Now turn on the big smile.
"Perfect. I'll be over for dinner at 7:30... no, 8. And I'll bring the wine this time... you just cook that special thing you made for us last week.

Let her reply. If she smiles and plays along, introduce yourself.
Tested and approved.
Here's another one. I want to see some replies with your ideas as well! Let's make the supermarket the new disco, baby! Yeah baby!
"My buddy is having a girl over for dinner. What should he make?"
"Well, girls always like pasta..."
"Actually, I didn't need to know that. I just made that up so I could come over here and flirt with you. Did you not get that? Here, let me try again.
How you doin'?"
