Officers from the force's Scotland division carried out a "day of action" targeting offenders who had allegedly committed serious offences on the rail network.
Among those arrested was a 14-year-old boy allegedly found with a knife.
Officers also arrested a 52-year-old man who allegedly used racially aggravated language on a train travelling between Glasgow and Cumbernauld.
Two boys, both 12, who were said to have spat at a ticket examiner on a train in the Greenock area were also arrested, as was a 19-year-old woman who allegedly used sectarian language on a Glasgow Central-Kilmarnock service.
All those arrested are subject of reports to the Procurator Fiscal and will appear in court in due course.
Others arrested included passengers who allegedly tendered counterfeit money to pay their fares.
Detective Inspector David Gray, of British Transport Police (BTP), who co-ordinated the day's operations, said: "Crime remains low on our rail network.
"However, when an offence does takes place we investigate it fully and, as the day of action results show, arrest those responsible and bring them before the courts."
Meanwhile, the Police Investigations & Review Commissioner has found a BTP officer was justified in using a spray to quell unruly fans after a Celtic v Ajax Champions League match in October.
In his report published today, Professor John McNeill said the two officers, who were outnumbered by fighting fans, had handled a violent situation professionally and that the use of what is known as Captor Incapacitant Spray was proportionate.