Officers are unsure if the whole phenomenon is a prank, a genuine mystery or a case of mass hysteria.
A victim named only as Sapna, 20, from the village of Devi Lal Nagar, claimed she was asleep beside her children, when she saw a cat enter her room. Sapna claimed that the cat transformed into a woman who pounced on her and cut her locks, reported the Times of India.
Several media outlets also reported a 28-year-old woman, Reena Devi, making similar claims.
While some of the details seem absurd, in a country where superstition and panic can spread easily, the tale has a darker side.
A "low-caste" woman was allegedly beaten to death by a group in a village near Agra, around 200km north of Delhi, on Wednesday and police say the attack may have been spurred by claims of witchcraft against the woman, but cautioned against spreading false rumours.
The 65-year-old woman - known only as Mandevi - was allegedly beaten to death in Mutnai, with police immediately launching a manhunt for two brothers, named as Manish and Song.
Superintendent Dinesh Chandra Dubey, of Agra police, said: "We will take action against people who are circulating a false story that the elderly woman was murdered because people suspected her to be a ghost who chopped off women's braids."
Villagers along the border between the state of Haryana and the National Capital Region have been putting up religious symbols to ward off "evil spirits", as rumours spread that the hair theft is linked to supernatural forces.
Some have even sent female relatives away from the region for safety.
Several academics and commentators have described the events as a case of mass hysteria, similar to incidents in 2001 when people reported being attacked by a "monkey man" - claims that turned out to be false.
Similarly, in 2006, hundreds travelled to coastal areas to sample seawater that suddenly "tasted sweet".
Forensic teams have visited the places where attacks are said to have occurred, but police are genuinely worried about a mob mentality taking hold, especially in rural areas where lower caste and mentally ill people are especially vulnerable to baseless accusations and tribal vendettas.
Deputy Superintendent Durgesh Kumar Singh, of Ghaziabad police, said: "We have instructed all our police staff to take care that such incidents don't create a panic.
"We have also asked them to speak to people so that they do not get carried away after hearing about the incident."