The song playing when the infant Stooges are dropped at the orphanage is "Road Runner" by Jonathan Richman and the Modern Lovers. Richman is one of the street guitarists in There's Something About Mary (1998), also directed by Bobby Farrelly and Peter Farrelly.
During the 'dentist' scene early in the movie, the three boys claim to be 'dabbling in the arts' when Mother Superior comes into the room to summon them to lunch. Two of the three are doing things that mirror interests in their lives. Moe is reading something (according to his brother Jack) he loved to do. Larry is playing the violin which he was been quite proficient on (initially the violin training was in response to a childhood accident), indeed good enough his parents seriously considered sending Larry to Europe to study at a music conservatory.
Jerod Mayo and Troy Brown of the New England Patriots play the two "gang bangers" whose pants are falling down. The Farrelly Brothers often include New England sports stars in their movies.
When Larry and Curley reunite with Moe on the Jersey Shore set, the producer who got Moe on the show is surprised to see there are three of them, conveniently forgetting he was watching all three together on stage before Larry and Curley walked off and he gave Moe the part.
When the Stooges are in the hospital's baby room the clock is at twenty before four but when they are being chased by the police a few minutes later it is twenty after three.
Peter and Bobby Farrelly say they think the Three Stooges are the funniest guys who ever lived and wanted to honor their work. Instead of taking material from the original Stooges, they decided to come up with their own new material.