The Wide Screen VHS version released by BMG contains all the cut scenes in their complete and uncut version.
Regarding Alternate versions in MCA/Universal laserdiscs and videos: as recently as the 1985 edition all missing scenes were restored to the laserdisc. It was the first Discovision pressing in 1981 which removed the previously listed scenes, and this was because early laserdiscs were not capable of quite as high capacity as in later years. The studio removed those scenes for the laserdisc because they were not essential to the plot and they allowed the movie to fit onto the Discovision disc. (HBO, for instance, ran Flash Gordon at the same time that Discovision released their edition but HBO included the missing scenes.) Later editions which continue to delete those few scenes are unremastered versions made from the Discovision edit.
For some reason, the most recent VHS release does not include the scene with Dale fighting Ming's guards in the hallways. The scene does appear on the DVD though.
In some versions, Ming's face is not visible in the scene where the pilots are sucked out of the plane. It is just the red clouds coming toward the plane.
In some versions that air on network and cable Richard O'Brien's scenes as Fico are cut - Flash and Princess Aura arrive, watch the tree-stump ceremony, witness the boy ask to be killed, and then they cut directly to Flash undergoing the same ceremony.
In the Universal Laserdisc and Video versions, the scene where Flash suggests using the curtains as parachutes while Zarkov flags down Ming's shuttle, as well as the scene where the young boy in Prince Barin's tribe asks to undergo the woodbeast test (complete with the shots of the beast inside) are cut. Also missing is Dale's fight with the guards & Ming's face from the meteor that attacks the plane. These scenes are restored on the Image DVD version.
Most versions also don't have the scene where Klytus wakes up on his ship and is informed that Barin and Flash are Hawkmen prisoners. The Image DVD restores this scene as well.
The 1993 Universal Laser Disc is complete with all deleted scenes listed in the other comments present on that disc, as well as the trailer being added at the end of the film. Reports from people who saw the film's preview in early 1980 indicate that there were several additional scenes of Flash on Aboria with Prince Barin.