Starting the conversation
The best benefit of approaching a lady in a supermarket is that the location provides you an endless list of low key conversation starters. If you like food, you will have lots to talk about. However, if you know nothing about cooking this will also serve you well, as many women learn how to cook and you will be understanding and ready to help a clueless bachelor. Break the ice by asking her for advice, or ask her for her opinions on how one food item would taste versus another. This gives you the chance to allow her the chance to talk about something she may love, while leaving room for you to compliment her on her extensive knowledge. And don't forget that conversations about meals can easily segue into conversations about weekend plans, which can easily turn into setups for asking her out.
Keeping the conversation going
This part can be tricky, but is vital if you're looking to get a telephone number and interest in a date. It's tricky because someone you meet at the supermarket is generally not there to socialize, but rather to finish her errands and return home. So it's important to strike a balance between conversing with her long enough to create a connection and not hanging around so long that she gets annoyed. Try to move from your opener for you to get her name to asking her out as fast as possible without raising the tension of the conversation or appearing to be nervous. If you can't seem to make all of this happen before she needs to begin the next aisle, just let her go and see if you can�t run into her later, maybe near the checkout. Definitely don't follow her throughout the store, but look for a chance to meet her near the front of the store. Casually approach her and invite her to coffee or lunch sometime in the next couple of days.
Remember that the important thing to making all of this successful is keeping pressure low and the vibe casual. Fortunately you're not operating in the meat market that's the bar scene, but rather a real meat market in the grocery store. A lighter mood can relieve a lot of the pressure normally associated with approaching a woman and can improve your chances for success.