What are some specific examples of superstitions of the Puritans in The Crucible?
Because the Puritans believed in the Devil and witchcraft, they believed that someone could have the ability to suss out both. Reverend Parris sends for Mr. Hale from Beverly to come to Salem and investigate because he has "much experience in the demonic arts." Such a belief in these arts is superstitious. Ironically, Mr. Hale later says, "We cannot look to superstition in this"; however, it is exactly superstition that makes a belief in his experience and powers possible.
The Putnams send their daughter, Ruth, to Tituba -- Reverend Parris's slave -- to "conjure up" the spirits of Mrs. Putnam's dead children. The Putnams superstitiously believe that "They were murdered" by a witch and that now, "some power of darkness [...] stop[s]" their one surviving daughter's mouth, making her ill and unable to speak.
Another superstitious belief is revealed when Betty accuses Abigail of drinking blood as a part of a "charm [meant] to kill Goody Proctor." This belief in the ability to cast spells accompanies the belief in witchcraft. Mr. Hale is especially interested in the fact that the girls were dancing in the forest, in the fact that there was a pot on the fire that night, and that there was something alive that had been cast into that pot.
The Puritans believed that the forest was the home of the Devil. It contained mystery and the unknown so they presumed it to be pure evil.
...the Salem folk believed that the virgin forest was the Devil's last preserve, his home base and the citadel of his final stand. (Act I, beginning narration)
The Puritans believed so firmly in the Devil's existence, that they believed his children (like Tituba) could speak to the dead:
Mrs. Putnam: Tituba knows how to speak to the dead, Mr. Parris. (Act I)
The poppet cited regularly in Act III that Mary Warren made for Mrs. Proctor must have been believed to be like a voodoo doll as Abby played the part of being struck by it.
The Puritans also believed in spectral evidence. That is to say that ghosts or witches had the power to intervene in human lives and that can be discerned by people. The entire play is about the children pretending that they are being haunted by various members of the church. This pretend ends the lives of many people.