Since the small ships requisitioned for the evacuation of Dunkirk were selected by the Royal Navy or a branch of the Admiralty, there must be a list of vessels so requisitioned. I ask for this information since various figures are given from 700 small ships to 1300. How can we find the truth?
Immediately after the Dunkirk evacuation was completed, Lt. Col. G P Orde compiled a list of all the vessels involved, using all available sources (remember that some of the 'Little Ships' were privately owned vessels that were not officially requisitioned, but which had had just "followed the fleet" to help out with the rescue). Even so, he acknowledged that there may still have been omissions.
Many of the original sources that Lt. Col. Orde consulted were subsequently lost or destroyed, so that list is generally considered to be the most authoritative, and is the primary source on which subsequent lists have been based.
The list is often referred to as The Dunkirk List or The Orde List (officially: Dunkirk Withdrawal: Operation Dynamo May 26–June 4, 1940: Alphabetical List of Vessels Taking Part, With Their Services. 940.542.1'1940). I don't think it has ever been published online, but the original copies are available to consult at the Caird Library at the National Maritime Museum.
The National Maritime Museum in Greenwich, London has published a research guide to the Dunkirk List which you might find helpful.
Subsequent research has added to the Orde list. Notably, Richard Collier identified some further examples which were included in the list in his 1961 book The Sands of Dunkirk, as did Walter Lord for the list in his 1981 book The Miracle of Dunkirk. Most recently, Russell Plummer has been able to add yet more names to the list included in his 1991 book The Little Ships that Saved An Army.