In which year did Dumbledore become the Headmaster for Hogwarts school of Witchcraft and Wizardry?
Although the generally accepted date is 1956 (e.g. presuming that Minerva McGonagall replaced Dumbledore as Transfiguration Mistress on his appointment to Headmaster) there's a fair argument that he didn't actually become Headmaster until late 1964 / early 1965.
You can see an extensive discussion of this alternate hypothesis here
Though never specifically stated, we can infer that Dumbledore became Headmaster in 1956. How do we know this?
Prior to being Headmaster, he served as Transfiguration professor under Headmaster Armando Dippet. Thus, when Dumbledore succeeded him he would have left a vacancy for the spot of Transfiguration. We know that Minerva McGonagall filled this vacancy, but the question is when?
From Order of the Phoenix, Chapter 15: The Hogwarts High Inquisitor:
Order of the Phoenix takes place in 1995 and 1996, with this statement coming in approximately September 1995.
Given the odd date of hiring (in the middle of the school year), we can infer that Dippet died in office, Dumbledore succeeded him, and McGonagall hired to replace him as soon as possible.
Additional Evidence for 1956
We also know that Tom Riddle jynxed the position of Defense Against the Dark Arts when Dumbledore (as Headmaster) refused him the position. Again, this is another date that is unconfirmed but is generally accepted as circa 1957.
Unfortunately this is based on the ambiguous