According to The I-Spyder Book of Earth Creatures, the sabre-toothed tiger — formally known as Smilodon fatalis — was a species of cat that died out around 9000 BC because the global climate change killed off its prey. It had two remarkably-sized upper canine teeth, measuring up to 17 cm in length. It was possessed of a stout, muscular body with a thick chest, and retractable claws. It primarily ate herbivores like bison.
The last sabre-toothed tiger from Earth was saved and kept in the Museum of the Last Ones. This specimen was cloned by Eve to distract people while the cloned dodos planted bombs. The sabre-toothed tiger was worth 500 points in the I-Spyder Book of Earth Creatures. Martha Jones spotted one, but only because she and the Tenth Doctor were being chased by it in England. They managed to lure it into some freshly-laid tarmac, whereupon the Doctor used the sonic screwdriver to set the asphalt and capture the tiger. (PROSE: The Last Dodo)
In an alternative timeline the sabre-toothed tiger was one of the creatures brought back from extinction by the Silurians. They were allowed to roam free and some were possibly used as mounts by the Silurians. (PROSE: Blood Heat)
Professor Payne brought a sabre-toothed tiger to 1890s London from thousands of years in the past to act as a guard. Leela broke its neck but sustained several injuries in doing so. (AUDIO: Chronoclasm)
On other worlds
K9 once encountered some sabre-toothed tigers on an unnamed planet in another dimension, partially inhabited by ex-Tellac miners. They lived alongside other Earth animals from widely varying periods of evolutionary history. On this planet, he observed one fighting a mammoth. (PROSE: K9 and the Missing Planet)