The Patiala State Monorail Tramway was a single rail laid next to an existing roadway. Four steam locomotives were ordered from Orienstein and Koppel of Germany (#3356-3359/1909) with an 0-3-0 wheel arrangement; a large waggon type wheel running on the roadway kept it from falling over in an arrangment called the Ewing System.
A single example along with a carriage are restored and operational at the Indian National Railway Museum in Delhi:
[www.dself.dsl.pipex.com]
There is another unrestored example as well at the NE Railway works, Amritsar. The railway ran from 1907 to 1927.
Otherwise, the honor is usually given to the Romney, Hythe & Dymchurch Railway and the Ravenglass & Eskdale Railway. I think the Bure Valley Railway is more of amusement park railway; does it run from point A to point B, and provide a passenger service?