Kam G. Salgueiro
This answer contains spoilers… (view spoiler)[There is the Palais Garnier, which inspired the setting of the book (the Opera House really has an underground lake), and different events that happened there, such as the falling of the chandelier, really occurred (this killed a woman). Even some characters are thought to be inspired from events, such as Erik, who was a man that worked on the construction of the Opera and fell in love with a singer, the singer rejected him and he kidnapped her for a few weeks. After letting her go, it is said that he went underground and remained there until he died of starvation below the Opera, were the real lake and were he lived. (hide spoiler)]
Sylvia Bloom To add to Kam's answer, Christine is believed to have been inspired by Christine Nilsson, a nineteenth-century opera star. Both are Swedish, and Leroux's descriptions of Christine in the book fit Christine Nilsson's appearance to a tee.
Julia The Phantom This story is fictional, but was inspired by a chandelier counterweight falling on a unfortunate audience member during a show. Many blamed it on a ghost in the Palais Ganeir. That was Leroux's first inspiration for the novel.
Emileine I wouldn't venture to claim that I could give anymore concrete evidence that The Phantom of the Opera was based on a true story than has already been given (ie, the chandelier did fall and a woman was killed; there was an oddly placed skeleton under the opera house by a subterranean lake, etc.). However, after reading the prologue to this book by Gaston Leroux and learning about bit more about him, I think this was definitely based on a true story, if he simply did not change the names to protect the innocent. He specifically recounts real life people he had spoken with (such as Meg Giry and the Persian) and stated that while ransacking the archives of the National Academy of Music, he came across several references to "the phantom." It gives food for the thought :) If you have a chance to read this prologue, I think you would find it very insightful.
Heather Culley Furthermore, Shah, Raj (2014). "No Ordinary Skeleton: Unmasking the Secret Source of Le Fantôme de l'Opéra". Forum for Modern Language Studies. 50 (1): 16–29 (17; 25n11). doi:10.1093/fmls/cqt048 This article reveals, for the first time, how the manuscript explicitly cites a hitherto unexamined inspiration for the novel, from which Leroux derived his central plot device of the eponymous Phantom's skeleton. This fictional cadaver has its roots in an apparently historical anecdote concerning the use of a real skeleton in performances of the Paris Opéra's 1841 production of Le Freischütz.