A little unfairly, the spirit that haunts this old edifice is nicknamed "Freddy" after Freddy Krueger (of "A Nightmare on Elm Street" fame,). Freddy has made his presence known to curators, staff and even a film crew in his (her?) time.
Doors open and close, electrical equipment sometimes unplugs by itself and the feeling of a "chill" or being watched has been reported in the building. Most common (from our own reports and reading Barbara Smith's Ontario Ghost Stories and John Robert Colombo's Mysteries of Ontario) are odd sounds heard by staff late at night.
The theories are put forth that the ghost could be the spirit of anyone from a Cree warrior who's clothing is displayed to Sir Wilfrid Laurier but more likely than not, according to Ms. Smith, it's the building's architect who killed himself jumping from the roof of the building.
We have had one person write to us that the architect did not commit suicide in the building (or at all,) but many other sources say that he has.
Our Ottawa team is trying to validate this part of the legend.
We would strongly recommend those interested in this read the above mentioned books or take the ghost tour offered in Ottawa by Haunted Walks of Canada and ask your guide about the museum.