The 73-year-old victim was left shaken by the incident near the junction of Hillylaid Road and Lawsons Road, at around 1am on Sunday.
Det Con Mick Englishby of Fleetwood CID said: "The lady had been to Thornton Sports and Social Club and was walking home with her partner in the early hours.
"A young male tried to snatch her bag, but she resisted and he ran away without getting anything.
"But it is worrying that this sort of thing is happening.
"The lady suffered a scratch to her leg and she was badly shaken up."
Now officers would like to hear from anyone who may know the culprit.
But this was such a hard game with Leek Town, desperate for points, and an equally taxing pitch making life very difficult for Fleetwood as they battled to take the vital three points. In a night of heroes, Danny Hurst deserves a special mention for his fearless goalkeeping, together with Steve Foster and Warren Beattie for two cracking goals.
He is described as aged 18 to 20 with a thin face and short dark hair and wearing dark clothing.
Anyone with information should call Fleetwood police on (01253) 876611 or Crimestoppers on 0800 555 111.