After an incident in which Nyssa demonstrated a knowledge of the algorithmic principles of fruit machines, the Fifth Doctor told her to remind him never to take her to Las Vegas - a sentiment Turlough disagreed with, preferring the opposite. (AUDIO: Eldrad Must Die!)
Around 1 December 1955, Frank Sinatra was playing in Las Vegas. He borrowed Elvis Presley's mobile phone around this time and because of this the Thirteenth Doctor was able to arrange for Elias Griffin Jr and his wife to get VIP access to Sinatra's show. (TV: Rosa)
In 1960, the Eighth Doctor dropped off Fitz Kreiner and Flossie in Vegas to investigate Brenda Soobie. They found Brenda at the Hotel Miramar and encountered agents of MIAOW. Fitz, Flossie, Brenda, and Noël Coward were captured by Mida Slike and taken to dogworld. After Margaret was defeated on dogworld, Coward used the Blithe Pinking Shears to bring Brenda, Flossie, the Doctor, Fitz, Anji Kapoor, and Fritter to Vegas. (PROSE: Mad Dogs and Englishmen)
The Tenth Doctor and Heather McCrimmon visited Deep Ocean, a huge, Luxembourg-sized underground biosphere that had been constructed in the city. There, the duo encountered a Thalatth, a species capable of extraordinary metamorphosis. (COMIC: Hook, Line and Sinker)
Behind the scenes
Though it's possible that a real world location doesn't exist in the same geographic space in the Doctor Who universe, such cases are few and far between. Thus, the map at right is probably a good indicator of the DWU location of Las Vegas.