I know what you're thinking. How can this campy musical ever show signs of Sigmund Freud. Well it actually does. Through it's romantic female lead Sandy to the T-Birds ways the movie actually shows a lot of Freudian sub plot.
-Sandy herself is the first one. Obviously representing the Super-ego of the time. With her dress and conservativeness she is a prime example of anti-Id. But in the end to experience her pleasure she changes who she is and becomes Id itself. While experiencing her Genital stage in psychosexual development. Which converts her to the Pink Lady Id-ness that she soon finds to love. She also goes through the repression of the unconscious by repressing her old self to fit a situation with her attitudes fighting just to create some significance of self-gratification.
-Another great example of the Freudian ways is Danny's amazing Ego. The fact that he starts as an Id and changes his ways to fit Super-ego and balancing it out with his Ego to try and satisfy one woman. He also is going through the final stage of psychosexual development with his symbolic gratification of Sandy. Seeing that he wants her (Oedipus complex changed to create an illusion as head of a family) as his girlfriend no matter what, creating the gratification.
-Also the two groups (Pink Ladies and the T-Birds) are great examples of the drives. They both are wanting of the Life drive and experiencing life to it's fullest even if it could hurt them. With there anti-culture of smoking and sex, not to mention the street racing and boozing it up from the T-birds.
But near the end of the movie they all seem to balance it a bit more with there Egos and find a happy medium by going back to school, stopping bad habits or changing themselves with in a boundary.