Continuity mistake: Whilst Fry, Leela and the professor are on the log near the waterfall they are very close to the right side of the river. A few shots after Gunter puts the hat back on his head you can see they and the log are now all the way over to the left side of the river.
Continuity mistake: After the accident when Fry looks at his body, there is a tool kit or something similar more or less right in between the body's legs. In a few moments there is a shot from above and the kit is several feet to the right of the body without anyone touching it.
Continuity mistake: When the hippie tells everyone to join hands, the cord from the microphone change position in the next shot. (00:19:10)
Continuity mistake: In the intro, before the main titles, the Professor informs the crew that he's booked them on the maiden voyage of the Titanic. When Leela says "Looks nice," at the beginning of the "scary" Titanic montage, the right side of her shirt is undamaged. During the middle of the "scary" Titanic montage, it suddenly becomes torn.