With a new baby in the house, I've spent many minutes over the past week trying to coax a precious smile from my 4 week old son! We've seen the occasional smile over the past 2 weeks, but we've had to work hard for them! I've joked that it's easy to distinguish between that true social smile and a 'gas' smile because the 'gas' smile appears suddenly and disappears just as quickly...you see the corner of baby's lips turn up briefly and drop suddenly, whereas, the social smile develops slowly (and usually takes a little tickling and encouragement from you to fully develop) and fades away just as slowly too.
New babies usually flash their first real smile around 4 weeks of age. This is their first true social interaction with you so it's an exciting milestone! Once you see that first smile, it might remain evasive for several more weeks, but baby should gradually start flashing you that smile frequently...and melting your heart every time! If your infant is not yet smiling, don't worry smiles can be easy to miss and a baby who doesn't smile a lot isn't necessarily headed for communication delays. The next milestone to watch for is the cooing and gooing (these noises are also heart-melting!).
Enjoy those smiles when you can get them!