Both incidents happened in the central Scotland area on Saturday evening.
In the first incident, two men racially abused a man and another couple at around 8.20pm on the 19.44 Glasgow Queen Street to Dunblane service.
British Transport Police said the men, who got off the train at Stirling, are both between 30 and 40-years-old, with very short hair and dark clothing.
Constable Ross Falconer said: "I would appeal for anyone who was on the train, at the stations, or has any information regarding the identity of those responsible for this unacceptable and unprovoked racial abuse, to contact me."
Shortly after 9pm the same evening a 46-year-old man was punched by a group of five men when he challenged them for chanting sectarian songs on the 20.18 Glasgow Queen Street to Alloa service.
The victim, who was travelling with his family, suffered a facial injury and a cut to his head but did not need to go to hospital.
The group got off the train at Stirling. One of the men is stocky with short brown hair and was wearing a khaki body warmer. A second man is around 50-years-old and was wearing dark clothes. A third man is about 5ft 10in and wore a quilted jacket.
Investigating officer Derek Jackson is appealing for information.
"The victim was subjected of a deplorable attack and I am anxious to trace those responsible as a matter of urgency. If you have any information please come forward," he said.