One man is in jail and another is awaiting sentence for racially abusing a French woman on a Melbourne bus, in violent, expletive-ridden tirades that gained international attention.

Hayden Stirling Stewart, 25, was taken into custody on Friday for his rant against Fanny Desaintjores and her friends while on a bus travelling between Mordialloc and Highett on November 11, 2012.

Fanny Desaintjores, inset, with David Graham, who abused her aboard the bus. Credit:YouTube

David Robert Mathew Graham, 36, also faces a jail term for what magistrate Jennifer Goldsborough described as the most vile, horrible and violent language she had heard in 17 years of sitting on the bench.

The tirades were captured on video with a passenger's mobile phone and gained international notoriety. The video had been viewed more than four million times since it was uploaded to YouTube, Moorabbin Magistrates Court heard.