(covers information from several alternate timelines)
In 2293, the location of Rigel in the Milky Way Galaxy was labeled in a star chart that was in Captain James T. Kirk's quarters aboard the USS Enterprise-A. (Star Trek VI: The Undiscovered Country, okudagram)
In 2366, in an alternate timeline, during the Federation-Klingon War, the location of Rigel was labeled on a tactical situation monitor in the ready room aboard the USS Enterprise-D. (TNG: "Yesterday's Enterprise", okudagram)
|Rigel II • Rigel III • Rigel IV • Rigel V • Rigel VI • Rigel VII • Rigel X • Rigel XII • An asteroid belt|
|See also: • Beta Rigel • Rigel Beta|
A cut line in the script of Star Trek IV: The Voyage Home had a Starfleet Command Table Operator report, "Rigel monitoring graphics general data synth with graphic reps on main screen 3," during the Whale Probe crisis of 2286. 
According to Star Trek: Star Charts (p. 36) and Stellar Cartography: The Starfleet Reference Library ("Federation Historical Highlights, 2161-2385"), the Star Trek universe had two Rigels. The first – Rigel (Beta Orionis) – was the "true" Rigel. This was a bright blue binary star, 773 light years from Sol. The second – Beta Rigel – was a system known for its high percentage of inhabited planets. Four of its planets were M-class. Both Rigels were located in the Beta Quadrant.
The "true" Rigel was located in the constellation Orion, which is visible on-screen in TNG: "Family" and on a constellation star chart in TOS: "The Corbomite Maneuver". The other Rigel was apparently not a star named by Humans, since Vulcans and Klingons used this name in their native languages and the location of Rigel was not known to Humans before it was learned from the Vulcan star charts in "Broken Bow".
According to The Worlds of the Federation (p. 36), Rigel and Orion were names for the same star.
According to Star Trek Maps (Chart D), the Rigel system was in Beta Orionis.
According to the Star Trek Encyclopedia (4th ed., vol. 2, p. 222), the Rigel referenced in Star Trek is the star Beta Orionis. This star is described by the authors as a "bright supergiant star, visible from Earth as one of the 'legs' in the constellation Orion (the hunter)."