A man, in hospital after being shot in the back, is refusing to speak to police, in one of three Sydney shootings on Australia Day.

The man was taken to Auburn Hospital in Sydney's west about 5pm yesterday with a gunshot wound to his back, but refused to give police a statement about what happened.

Two men in the car that took him to hospital were also interviewed, but investigators said they were not believed to have shot him. Police said they were also searching Perry Way, Auburn.

A 24-year-old man was shot twice in his leg at Bolton Street, Guildford, in western Sydney about 10.15pm, police said.

He was taken to Westmead Hospital but his injuries were not believed to be life-threatening, police said.

In the first incident, shots were fired into a tanning salon on the Princes Highway in Rockdale in southern Sydney about 2am.

No one was hurt but the bullets hit the front window and door.

Police are appealing for anyone with information to phone Crime Stoppers on 1800 333 000.