Lawyers representing Sergeant Danny Nightingale, 37, are due to ask appeal judges to analyse the case at a hearing in London.
Sgt Nightingale was sentenced to 18 months' military detention earlier this month by a judge sitting in a military court.
He admitted illegally possessing a Glock 9mm pistol and more than 300 rounds of ammunition at a court martial.
His wife Sally, 38, who lives near Crewe, said before the start of the appeal hearing in London that more than 106,000 people had signed a petition calling for him to be freed.
"The public interest has just overwhelmed us," said Mrs Nightingale, as she arrived at the Royal Courts of Justice, where the appeal will be heard. "It has helped me fight on."
Three judges - the Lord Chief Justice, Lord Judge, Mr Justice Fulford and Mr Justice Bean - are scheduled to hear legal argument at a hearing in the Court Martial Appeal Court.
Sgt Nightingale, a father of two, is being held in a military centre in Colchester, Essex.
He told the court martial in Bulford, Wiltshire, that the gun was a gift from Iraqi soldiers he had helped train.
The court martial heard that Sgt Nightingale, who has suffered medical problems affecting his memory, appeared to have put the gun in a cupboard in his army accommodation in Hereford on a "very hectic day" when preparing to deploy to Afghanistan. Sgt Nightingale said he had not "appreciated" that he had it.