A Casual Approach (Can Be Humorous)
This is the low-key approach. No need to apologize, no big deal. It should be used when some small amount of common experience has been established, such as when a girl smiles and gives some encouragement to you. Or when you are waiting together for something. Or, if you have a humorous observation to make to the girl about herself, you, or any situation nearby. It is also a good follow-up approach after a request for help.
Examples:
"Hey! Where're you going?" (Said in a teasing manner).
"Is it O.K. if we walk along together?"
"Say! If I bet someone that I could get you to stop and talk to me, would I lose my bet?"
"Do you mind if I ask you something?" (pause) "If I'm real nice, if I'm a perfect gentleman, would you let me walk along with you?" (Or sit with you).
"It looks like you are (or have) ..." (Then comment on some obvious thing about the girl in a casual way).
Any humorous opening line is effective in a casual situation. Joking references to shortcomings in public services, transportation systems or any other commonly known feature are ideal. A good joke about any subject will work. TV and radio are a good source for humorous material.