For some males, it is very difficult to strike up a conversation with a stranger, especially if that stranger happens to be of the opposite sex. Basically there are two ways to introduce yourself to a girl, directly or indirectly.
Direct Conversation Starters
Walk up to the target female. Smile and say, "Hi, my name is Mel. Can I buy you a drink?" This approach may catch her off guard, and she will accept the drink offer.
Compliment the target female on her choice of outfit, her jewelry or her hairstyle. Some girls like the special attention and will be open to future conversations.
Start a conversation about the surroundings or the weather. Chances are the target female will answer with pleasantries, opening the door for further conversations.
Indirect Conversation Starters
Observe the target female talking to an acquaintance. Walk up to the mutual acquaintance and start to participate in the conversation. The acquaintance will usually introduce the female, as good etiquette dictates.
Become a hero to the target female. Watch to see if the target female needs assistance, such as opening doors or carrying bags. Females may feel they need to repay the act of kindness with an evening at the movies.
Scope out the shopping mall. If there is an attractive sales clerk at a specific store, buy something from her. Go back the next day to buy something else. Ask her for assistance. A possible line to use would be, "I was here yesterday looking for a gift for my sister, but didn't have much luck. Could you help me find a present for a 14-year-old girl?" After she finds a gift, thank her by taking her out to lunch.
Borrow a puppy. Walk the dog near the targeted female's house or place of work. Stop outside and pet the dog. Chances are she will come out and comment on how cute the dog is. Take the opportunity as an ice-breaker and invite her on the walk.