Paul McManus, 19, is alleged to have killed Isabelle Sanders on April 9, at her home at Raeswood Drive, Crookston by repeatedly stabbing her on the head and body.
He is also charged with attempting to murder 85-year-old Norman Busby at the same house by stabbing him on the head and body to the danger of his life.
McManus, from Govan, faces a third charge of assaulting Daniel O'Donnell on April 8 at a car park near the Spar on Barrhead Road and striking him on the right leg with a knife to his severe injury, and attempting to rob him of his car.
The teenager appeared in private at Glasgow Sheriff Court where he made no plea or declaration and was remanded in custody by sheriff Stuart Reid.
He will appear at court again next week.