Locals in Weymouth, Dorset, where the victim lived in a housing association flat, suggested a 16-year-old had been arrested. Police said: "All parties were known to each other."
The single mother, named locally as Julie, was found dead at her home in Benville Road on Saturday after police received a report that she had been injured. Unconfirmed reports suggested she had been fatally stabbed.
Officers put a cordon in place around the three-storey block of flats while forensic experts examined the scene.
Armed police carried out door-to-door inquiries and inspected cars and properties in the neighbourhood while a police helicopter expanded the search. Dorset Police said they arrested the teenager on Saturday night.
The victim had lived in the flat with her son for about a year-and-a-half and continued to make frequent trips to Scotland to visit her mother.
Family liaison officers have spoken to her next of kin and relatives are travelling to England to identify the body. A post-mortem examination is being carried out.
Forensic examination of the scene was continuing last night. Two bouquets of flowers had been placed outside the flats.