The show boat "Cotton Blossom" is inaccurately designed for the era in which the story takes place (the 1880's). The boat used in the film is built in the style of a typical modern luxury riverboat, with giant twin smokestacks and a large paddlewheel in the rear, and it moves on its own power. Modern "show boats" are built that way because of advances made since the 19th century, but authentic show boats of the era did not have smokestacks or paddlewheels, and were not self-powered. They were barge-like structures similar to a long, floating house with a flat roof, and they were connected to, and pushed along by the misleadingly named "towboats", which did have smokestacks and a paddlewheel. If a real show boat of the era had been steam-powered, its steam engine would have had to be placed (very dangerously) smack in the middle of the auditorium.
In the "Ol' Man River" sequence, after the boat has stopped moving to allow Ravenal to climb on board, it begins moving again and passes some dock workers on rafts. Joe is seen in long shot, standing on the deck of the show boat and singing to them the words "You an' me, we sweat and strain...", etc. During that specific moment, his voice is not quite synchronized with his arm and lip movements (he shakes his fists for emphasis). Then, on the line "Ya get a little drunk", he is shown in medium shot, and his voice matches his lip movements perfectly.
At one point, Magnolia (Kathryn Grayson) refers to Lady Southweight and Hamilton Barsdale as being characters in "Tempest and Sunshine", a melodrama adapted from a then-popular novel. There are no such characters in that play. In the scene in which Cap'n Andy (Joe E. Brown) introduces the show boat actors to the crowd, Julie (Ava Gardner) and Steve (Robert Sterling) make another reference to Hamilton, and Cap'n Andy then says: "You have to see the play tonight folks, to learn their secret - "Tempest and Sunshine", beautiful drama of tears and laughter".
When the Cotton Blossom is pulling away from the dock, at the end of the movie, you can see big clouds of blue smoke pouring out the right side of the ship (near the rear). These are definitely exhaust gases from either a gas or diesel engine that is installed in the ship, and most likely used to power the paddle wheel.