What rumours have spread about Betty in The Crucible, and who is responsible for the rumours?
It actually isn't Betty, Reverend Parris's young daughter, about whom rumors are spread; it is Abigail, his seventeen year-old "strikingly beautiful niece." Parris asks her, in Act One, if her name "is entirely white [in the town]." What he means by this is that he's heard that she is not known to be entirely innocent and pure, as an unmarried Puritan woman ought to be. Further, he says,
Abigail, is there any other cause than you have told me, for your being discharged from Goody Proctor's service? I have heard it said, and I tell you as I heard it, that she comes so rarely to the church this year for she will not sit so close to something soiled.
In other words, it sounds as though Elizabeth Proctor, the wife of Abigail's former lover, John Proctor, or someone who suspects what occurred between Abigail and John is talking. It seems unlikely to me that it is Elizabeth spreading rumors about Abigail based on what we learn of her character later. She wouldn't wish to soil or tarnish her own husband's name by speaking of his infidelity. However, someone must have some suspicion about Abigail and John's affair because, as Parris says, it's been seven months since Elizabeth Proctor fired Abigail and no other family has inquired about hiring her. This person assumes that Elizabeth comes rarely to church because she wouldn't want to be near someone as morally corrupt as Abigail. However, we learn, in Act Two, that Elizabeth has rarely been to church this year because she's been ill. Later, in Act Three, we learn that Elizabeth Proctor never lies; her husband testifies that she cannot lie when he is in the courtroom speaking to Danforth. Therefore, it seems most likely that it is someone else -- not Elizabeth -- who spreads the rumors about Abigail.
But, of course, Abigail blames Elizabeth for the rumors. She says, "My name is good in the village! I will not have it said my name is soiled! Goody Proctor is a gossiping liar!" She has strong incentive to hate Elizabeth, the woman who is married to the man she loves, and the woman who dismissed her from their service, thereby separating her from him. Further, Abigail is so vitriolic and the evidence is stacked against her, so we can discern, even at this early stage, that she is lying about her name (and also, likely, who is doing the rumor-spreading).
My apologies! There is a rumor about Betty early on, in the beginning of Act One. When the Putnams arrive, Mrs. Putnam asks how high Betty flew, and the Reverend Parris denies that Betty ever did fly. However, Mrs. Putnam says, "Why, it's sure she did; Mister Collins saw her goin' over Ingersoll's barn, and come down light as bird, he says!" So the rumor going around is that Betty's been witched and that she has actually flown as a result of it. It is unclear exactly who is responsible for this rumor, as we learn from Abigail that the whole town is talking about witchcraft. Presumably, Mr. Collins started the rumor since he's the one who described the location of Betty's flight and her landing, but we never meet Mr. Collins or hear anyone verify that the story began with him.