Junior Bradshaw, 30, and Kevin Liverpool, 33, were arrested Monday near Stone's house in Cullompton, a city 175 miles southwest of London, after residents reported a suspicious-looking vehicle.
They appeared in court Thursday in southwest England on charges of conspiracy to rob and conspiracy to commit grievous bodily harm.
The prosecutor told the court that items including the sword and notes relating to Stone had been found in the suspects' car.
The defendants spoke only to confirm their identities and were ordered detained until their next hearing on July 1. No members of Stone's family were in court.
The 24-year-old singer, who is due to start a European tour next month, said in a statement that she was "absolutely fine and getting on with life as normal."
The Associated Press contributed to this report.